SUBSURFACE STRESSES IN ANISOTROPIC CYLINDRICAL CONTACTS By SANGEET SUBHASH SRIVASTAVA
SUBSURFACE STRESSES IN ANISOTROPIC CYLINDRICAL CONTACTS
By
SANGEET SUBHASH SRIVASTAVA
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2003
Copyright 2003
By
Sangeet Subhash Srivastava
ACKNOWLEDGMENTS
I am grateful to Dr. Nagaraj K. Arakere, my committee chairman, who has given
me full guidance throughout this project. Also, I am thankful to the members of my thesis
committee, Dr. Ashok V. Kumar and Dr. John K. Schueller, for the time they spent in
reviewing and commenting on this study.
Special thanks go to my family members and friends for giving me support and
help throughout the study. Finally, I would like to thank the Almighty God for giving me
power and courage to complete this thesis.
iii
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ................................................................................................. iii
LIST OF TABLES............................................................................................................. vi
LIST OF FIGURES .......................................................................................................... vii
ABSTRACT.........................................................................................................................x
CHAPTER
1 INTRODUCTION ............................................................................................................1
History................................................................................................................................. 1
The Objective of This Study ............................................................................................... 3
Motivation for the Problem................................................................................................. 6
Isotropy ............................................................................................................................... 7
Anisotropy........................................................................................................................... 8
2 LITERATURE REVIEW AND THEORETICAL FRAMEWORK ..............................10
Hertz Theory of Elastic Contact ....................................................................................... 10
A Review of Articles on Plane Contact Problems ............................................................ 14
Contact Problems Neglecting Forces of Friction ................................................... 14
Contact Problems for the Case When the Punches are Rigidly Connected to the
Elastic Body .............................................................................................. 16
Contact Problems in the Theory of Elasticity Considering Friction ...................... 17
Problems of Contact Between Two Elastic Bodies................................................ 20
A Review of Articles on Three-Dimensional Contact Problems...................................... 22
Examples of Numerical Approaches to Contact Mechanics Problems ............................ 25
An Overview of Lekhnitskii’s Work ................................................................................ 27
3 ANISOTROPIC ELASTICITY EQUATIONS FOR FCC SINGLE CRYSTALS ........28
Slip Activation and Deformation in FCC Single Crystal.................................................. 29
Elasticity ........................................................................................................................... 31
Elasticity Matrix................................................................................................................ 32
Coordinate System Transformation .................................................................................. 35
Steps in Coordinate Transformation ................................................................................. 37
Stress Transformation ....................................................................................................... 40
Transformation of the Elastic Constants........................................................................... 41
iv
Slip System Shear Stresses ............................................................................................... 42
Resolved Shear Stresses.................................................................................................... 42
4 GENERALIZED PLANE STRAIN EQUATIONS FOR A HOMOGENEOUS
ANISOTROPIC ELASTIC SOLID...................................................................................43
Plane Strain Conditions for Isotropic Contacts................................................................. 43
Generalized Plane Strain Conditions for Anisotropic Contact Problems ......................... 45
Generalized Plane Strain Equations.................................................................................. 47
The Distribution of Stresses in an Elastic Half-Space Under the Influence of Stresses
Applied to the Bounding Plane ............................................................................. 52
5 FINITE ELEMENT SOLUTION OF THE SUBSURFACE STRESSES......................55
Finite Element Model ....................................................................................................... 55
Material Properties and Model Characteristics................................................................. 58
6 RESULTS AND DISCUSSION .....................................................................................62
Crystallographic Orientation 1.......................................................................................... 63
Crystallographic Orientation 2.......................................................................................... 68
Crystallographic Orientation 3.......................................................................................... 73
Crystallographic Orientation 4.......................................................................................... 79
Crystallographic Orientation 5.......................................................................................... 85
Comparison of FEM Solution with the Analytical Results for Case 1 Neglecting
Tangential Traction............................................................................................... 92
Comparison of FEM Solution with the Analytical Results .............................................. 96
Case1 ................................................................................................................... 96
Case 4 ................................................................................................................. 102
7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK.................109
Conclusions..................................................................................................................... 109
Recommendations for Future Work................................................................................ 110
APPENDIX
SAMPLE COORDINATE TRANSFORMATION WITH ACCURACY CHECKS .....112
Sample Coordinate Transformation................................................................................ 112
Checks for Accuracy....................................................................................................... 113
LIST OF REFERENCES.................................................................................................114
BIOGRAPHICAL SKETCH ...........................................................................................118
v
LIST OF TABLES
Table
page
3-1 Atomic densities on FCC crystal planes......................................................................32
3-2 Symmetry in various crystal structures. ......................................................................33
3-3 Direction Cosines ........................................................................................................40
6-1 Orientation 1 activated slip system sectors..................................................................67
6-2 Orientation 2 activated slip system sectors..................................................................72
6-3 Orientation 3 activation slip system sectors ................................................................77
6-4 Orientation 4 activated slip system sectors..................................................................83
6-5 Orientation 5 activated slip system sectors..................................................................89
vi
LIST OF FIGURES
Figure
page
1-1 Isotropic cylindrical indenter in contact with a flat isotropic body...............................4
1-2 Isotropic cylindrical indenter in contact with a flat anisotropic body ...........................5
1-3 Pressure profile due to the indenter on the flat body (a stress problem) .......................6
1-4 Blade orientation............................................................................................................7
2-1 Two bodies in Hertzian Contact ..................................................................................11
3-1 Orientation-dependant stress-strain behavior ..............................................................29
3-2 Load and slip directions and angles.............................................................................30
3-3 Primary resolved shear stress planes and directions....................................................30
3-4 FCC crystal structure ...................................................................................................31
3-5 Material (xo, yo, zo) and specimen (x″, y″, z″) coordinate systems. ..........................37
3-6 First rotation of the material coordinate axes about the zo-axis..................................37
3-7 Second rotation about the y-axis. ................................................................................38
3-8 Third rotation about the x′-axis. ..................................................................................39
4-1 Isotropic elastic half-space ..........................................................................................44
4-2 Distribution of forces in an anisotropic half-space......................................................46
4-3 Homogeneous Anisotropic Body bounded by a cylindrical surface ...........................48
4-4 Elastic equilibrium of an infinite homogeneous body with a parabolic profile ..........53
5-1 Crystallographic Orientations of the anisotropic body................................................56
5-2 Global and material coordinate systems......................................................................57
5-3 Isotropic cylindrical indenter and the anisotropic substrate in contact .......................58
5-4 Dimensions of the anisotropic contact model..............................................................59
vii
5-5 ANSYS SOLID 45 element.........................................................................................60
5-6 Refined meshing in the contact zone...........................................................................60
5-7 Meshed anisotropic FCC single crystal substrate........................................................61
6-1 Schematic of the polar coordinate system used in the subsurface contact region.......62
6-2 Orientation 1 primary resolved shear stresses; r = 0.3*a.............................................64
6-3 Orientation 1 primary resolved shear stresses; r = 0.8*a.............................................65
6-4 Orientation 1 primary resolved shear stresses; r = 3*a................................................66
6-5 Orientation 2 primary resolved shear stresses; r = 0.3*a.............................................69
6-6 Orientation 2 primary resolved shear stresses; r = 0.8*a.............................................70
6-7 Orientation 2 primary resolved shear stresses; r = 3*a................................................71
6-8 Orientation 3 primary resolved shear stresses; r = 0.3*a.............................................74
6-9 Orientation 3 primary resolved shear stresses; r = 0.8*a.............................................75
6-10 Orientation 3 primary resolved shear stresses; r = 3*a..............................................76
6-11 Orientation 4 primary resolved shear stresses; r = 0.3*a...........................................80
6-12 Orientation 4 primary resolved shear stresses; r = 0.8*a...........................................81
6-13 Orientation 4 primary resolved shear stresses; r = 3*a..............................................82
6-14 Orientation 5 primary resolved shear stresses; r = 0.3*a...........................................86
6-15 Orientation 5 primary resolved shear stresses; r = 0.8*a...........................................87
6-16 Orientation 5 primary resolved shear stresses; r = 3*a..............................................88
6-17 Contour plot for sigma (x)--Analytical solution........................................................92
6-18 Contour plot for sigma (x)--FEM solution ................................................................92
6-19 Contour plot for sigma (y)--Analytical solution........................................................93
6-20 Contour plot for sigma (y)--FEM solution ................................................................93
6-21 Contour plot for tau (xy)--Analytical solution ..........................................................94
6-22 Contour plot for tau (xy)--FEM solution...................................................................94
viii
6-23 Contour plot for sigma (z)--Analytical solution ........................................................95
6-24 Contour plot for sigma (x)--FEM solution ................................................................95
6-25 Comparison of stresses in the X-direction for orientation 1......................................96
6-26 Comparison of stresses in the Y-direction for orientation 1......................................97
6-27 Comparison of stresses in the Z-direction for orientation 1 ......................................98
6-28 Comparison of shear stresses in the XY-direction for orientation 1 .........................99
6-29 Comparison of shear stresses in the YZ-direction for orientation 1........................100
6-30 Comparison of shear stresses in the XZ-direction for orientation 1........................101
6-31 Comparison of stresses in the X-direction for orientation 4....................................102
6-32 Comparison of stresses in the Y-direction for orientation 4....................................103
6-33 Comparison of stresses in the Z-direction for orientation 4 ....................................104
6-34 Comparison of shear stresses in the XY-direction for orientation 4 .......................105
6-35 Comparison of shear stresses in the YZ-direction for orientation 4........................106
6-36 Comparison of shear stresses in the XZ-direction for orientation 4........................107
A-1 Two step coordinate transformation .........................................................................113
ix
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
SUBSURFACE STRESSES IN ANISOTROPIC CYLINDRICAL CONTACTS
By
Sangeet Subhash Srivastava
May 2003
Chair: Dr. Nagaraj K Arakere
Major Department: Mechanical and Aerospace Engineering
This study deals with an analytical procedure for calculating subsurface stresses
in anisotropic cylindrical contacts. The procedure uses the method of complex potentials
with the assumptions of generalized plane strain at the contact. This assumption allows
for five independent stress components to be evaluated in the half space (σx, σy, τxy, τxz,
τzx). σz is a function of the five independent stress components. Stresses are not the
functions of z-coordinate. The generalized plane strain assumptions allow for modeling
the shear coupling present in anisotropic materials.
The analysis procedure developed is used to examine subsurface stresses in
contact between a cylindrical isotropic indenter and a single crystal anisotropic substrate.
Subsurface stresses are studied for various crystallographic orientations. The resolved
shear stresses on the twelve primary octahedral planes [(111) <110>] are computed in the
vicinity of the contact, to predict evaluation of slip systems in the contact region.
x
Subsurface stresses are also computed using FEA, for a few cases, for comparison
purposes. The analytical solution procedure is shown to be an efficient and accurate
procedure for computing subsurface stresses in anisotropic contacts.
xi
CHAPTER 1
INTRODUCTION
Contact Mechanics is a branch of engineering that deals with the explanations
about the nature of contact between solid bodies and its detailed study. It is concerned
primarily with the stresses and deformation that arise when the surfaces of two solid
bodies are brought into contact [1].
Contact problems can be classified into the following two categories:
Problems where one body is elastic and the other is rigid
Problems involving two elastic bodies
In the first class of problems, termed “punch” problems, the contact region is
known a-priori. In the second class, termed elastic contact problems, the contact region is
initially unknown and needs to be determined.
History
It may surprise those who venture into the field of contact mechanics that the
subject of contact mechanics was first started in 1882 with the publication by Heinrich
Hertz of his classical paper “On the Contact of Elastic Solids” [2]. At first glance it
appears as if the nature of the contact between the two elastic bodies has nothing to do
with the electricity. But Hertz realized that mathematics was the same and so founded the
field, which has retained a very small but loyal following over the past one hundred
years.
Hertz always wanted to be an engineer and so in 1877, at age 20, he traveled to
Munich for further studies in engineering. But he was really more interested in
1
2
mathematics, mechanics, and physics. Under his parents’ guidance he chose physics
course and found himself in Berlin a year later to study under Hermann von Helmholtz
and Gustav Kirchhoff. In October 1878, Hertz began attending Kirchhoff’s lectures and
came across an advertisement for a prize for solving a problem involving electricity. He
asked for permission from Helmholtz to research the matter and got a room for
conducting the experiments. He wrote his first paper, “Experiments to determine an upper
limit to the kinetic energy of an electric current,” and won the prize [1].
Next Hertz worked on “The distribution of electricity over the surface of moving
conductors,” which would become his doctoral thesis. This work impressed Helmholtz so
much that Hertz was awarded “Acuminis et doctrine specimen laudabile” with an added
“magna cum laude.” In 1880, he became an assistant to Helmholtz after which he became
interested in the phenomenon of Newton’s rings. It occurred to Hertz that, although much
was known about the optical phenomenon when two lenses were placed in contact, not
much was known about the deflection of the lenses at the point of contact. Hertz was
particularly concerned with the nature of the localized deformation and the distribution of
pressure between the contacting surfaces [1].
In January 1881, at the age of 24, Hertz presented his theory, which will be
discussed in detail in the later chapters of this study to the Berlin Physical Society. The
members of the audience were quick to perceive its technological importance and
persuaded him to publish a second paper in a technical journal that gave Hertz notoriety
in the technical circles. But the theory first appeared in the literature in the beginning of
the last century, stimulated by engineering developments on the railways, in marine
reduction gears and in the rolling contact bearing industry [1].
3
The Hertz theory is restricted to frictionless surfaces and perfectly elastic solids.
Progress in contact mechanics in the second half of the 20th century has been largely
associated with the removal of these restrictions. A proper analysis of friction at the
interface of bodies in contact has enabled elastic theory to be extended to both slipping
and rolling contact in a realistic way. At the same time development in theories of
plasticity and linear viscoelasticity have enabled the stresses and deformations at the
contact of inelastic bodies to be examined.
Surprisingly, there are a very few books written purely on contact mechanics. In
1953 the book by L. A. Galin [3], Contact Problems in the Theory of Elasticity, appeared
in Russian summarizing the pioneering work of N. I. Muskhelishvili in elastic contact
mechanics. In 1980, an updated treatment by G. M. L. Gladwell, Contact Problems in the
Classical Theory of Elasticity, [4] was published. These books excluded the rolling
contacts and are restricted to perfectly elastic solids. In 1985 Contact Mechanics by K. L.
Johnson [5], was published which aimed at providing an introduction to most aspects of
mechanics of contact of non-conforming surfaces. This book excluded the restrictions of
previous books. Recently, in the end of the 20th century and the beginning of the 21st
century a few more books have been published. But on the whole the subject of Contact
Mechanics after being started with the theory of Hertz was inactive for the first half of the
20th century and has become active in the latter half of the century.
The Objective of This Study
The objective of this study is to develop an analytical procedure (described in
detail in chapter 3 and 4 of this study) for evaluating the subsurface contact stresses
between an anisotropic (single crystal superalloy) elastic body (half-space) and an
isotropic elastic body (half-space) in contact. Using the stress results obtained from the
4
analytical approach, the resolved shear stresses (RSS) acting on the twelve primary slip
systems are calculated. The RSS values determine the dominant and the activated slip
planes as a function of radial and angular coordinates.
The analytical technique used in this study is adapted from Theory of Elasticity of
an Anisotropic Elastic Body, by S. G. Lekhnitskii [6]. Here, a Hertzian-type distribution
is selected to simulate contact. The solution for Hertzian contact problems of the type
described in Fig 1-1, where an isotropic cylindrical indenter is in contact with a flat
isotropic body, is described in many books such as Contact Mechanics by K.L. Johnson
[5]. For the assumption of plane strain to be justified, the thickness of the solid (in the Zdirection) should be large compared with the width (2a) of the contact region [5]. The
contact width ‘2a’ is generally very small compared to the length of the body for practical
problems; hence the plane strain conditions are valid. The loading profile and the shape
of the contact region are calculated under the plane strain conditions [5]. The subsurface
stresses can also be evaluated under these conditions, the formulas for which are given
[5] and discussed in Chapter 4.
Figure1-1 Isotropic cylindrical indenter in contact with a flat isotropic body
5
This thesis deals with evaluating the subsurface stresses for an isotropic
cylindrical indenter in contact with an anisotropic (single crystal superalloy) elastic
substrate (Fig 1-2). The problem of evaluating subsurface stresses for a general
anisotropic half-space with applied normal and tangential traction forces has been solved
by Lekhnitskii [6] using an analytical stress function approach. This solution is valid
under the assumption of plane strain.
In this study the analytical procedure outlined in Lekhnitskii [6] for a general
anisotropic elastic solid has been adapted for evaluating subsurface stresses for an FCC
single crystal substrate. The normal and tangential traction forces are obtained under the
assumptions of Hertzian cylindrical isotropic contact. It has been shown that normal
traction forces for anisotropic contacts are also parabolic in nature as in Hertzian
isotropic contacts. This simplification will allow the use of an analytical solution [6] of
an otherwise very complicated problem.
Figure1-2 Isotropic cylindrical indenter in contact with a flat anisotropic body
6
Figure 1-3 Pressure profile due to the indenter on the flat body (a stress problem)
The accuracy and results of the analytical approach are further verified in chapter
5 by numerical method using a commercial FEA program, ANSYS. The subsurface
stresses computed by analytical solution are verified using FEA for a few
crystallographic orientations of the substrate.
Motivation for the Problem
Contact between anisotropic (single crystal superalloy) and isotropic elastic
bodies are of practical interest in turbine blade applications in aircraft and rocket engines.
Single crystal turbine blades are used in high temperature applications. These blades are
attached to turbine disks, made of isotropic nickel-based materials, using dovetail joints.
In many instances the turbine blades also have friction damping devices that are tuned to
rub against the blade of certain frequencies, to add Coulomb damping. The vibratory
Hertzian subsurface contact stresses induced at the damped surfaces and blade
attachments initiate crystallographic fatigue cracks. The goal of this thesis is to develop
an analytical procedure for computing subsurface stresses at the contact of a cylindrical
anisotropic body and a single crystal anisotropic substrate.
7
Figure 1-4 Blade orientation Source: Modified from [7]
S.G.Lekhnitskii [6] offers a good detail of complex potentials and integral
transform methods. Large classes of problems in the plane theory of elasticity including
those problems, which are reducible to this theory, are treated by the method of complex
variables and other ingenious techniques. It is impossible to develop the close form
solution to these problems. Thus, an analytical solution is developed for a problem
involving a state of stress of a homogenous anisotropic body bounded by a cylindrical
surface. In this case, the stresses do not vary along the width.
Isotropy
An isotropic material has identical properties in all the directions at a point, any
plane being a plane of elastic symmetry (infinite planes of symmetry). Many
polycrystalline materials exhibit isotropy. Isotropy results when the crystal size is small
relative to the size of the sample provided nothing has acted to disturb the random
8
distribution of crystal orientations within the aggregate. Mechanical processing
operations such as cold rolling may contribute to minor anisotropy, which in practice is
often disregarded.
Such materials have only two independent variables or elastic constants in their
stiffness and compliance matrices. These two elastic constants are generally expressed as
the Young’s modulus, E and the Poisson’s ratio, ν. However the alternative elastic
constants bulk modulus, K and /or shear modulus, G can also be used and can be found
from E and ν by a set of equations, and vice versa. These equations are as follows:
G=
K
E
2⋅ ( 1 + ν )
(1.1)
E
3⋅ ( 1 − 2ν )
(1.2)
Anisotropy
E. I. Givargizov [8] describes anisotropy as the dependence of structure and
properties on direction in space. A non-isotropic or anisotropic solid displays directiondependant properties. For example, a solid exhibits greater strength in a direction parallel
with the fiber (grain) than perpendicular to the fiber (grain). It is a result of the discrete
nature of the crystal lattice, and it is the property that distinguishes the crystalline state
from another solid state of matter, the amorphous.
Single crystal also displays pronounced anisotropy, manifesting different
properties along various crystallographic directions. Some polycrystalline materials may
also exhibit anisotropy. Composite like wood is an example of anisotropy. Recent manmade materials that have joined wood are fiberglass, metal matrix fibrous composites
sandwich constructions, and fortisan composites. S.G. Lekhnitskii in his book [6] gives a
9
detailed study of anisotropic materials. He gives a few examples of anisotropic materials
of non-crystalline nature like pinewood, delta wood and plywood along with the
numerical values of their elastic constants.
Anisotropic materials play an important role in modern technology. Missile and
aircraft designers, specialists in mining problems, solid-state physicists, geophysicists,
manufacturers of certain parts and materials, and in general, people engaged in all
material sciences-all has to deal with a variety of anisotropic problems. Such materials
have 21 elastic constants in their stiffness and compliance matrices. A material without
any planes of symmetry is fully anisotropic whereas a material that has at least two
orthogonal planes of symmetry and where material properties are independent of
direction within each plane is orthotropic. Such materials require 9 elastic constants in
their constitutive matrices. Special classes of orthotropic materials are those that have the
same properties in one plane (e.g. the x-z plane) and different properties in the direction
normal to this plane (e.g. the y-axis). Such materials are called transverse isotropic, and
they are described by 5 independent elastic constants, instead of 9 for fully orthotropic.
CHAPTER 2
LITERATURE REVIEW AND THEORETICAL FRAMEWORK
A review of relevant literature in contact mechanics is covered in this chapter. It
also covers the detailed explanation of the introduction of contact mechanics by Hertz in
the late 19th century and all the assumptions he made in his theory. After contact
mechanics was introduced by Hertz in 1882, the field was surprisingly dormant in the
early first half of the 20th century and then research started again in the latter half. In the
last 70 years the field has gained pace.
Hertz Theory of Elastic Contact
The first satisfactory analysis of the stresses at the contact of two elastic solids is
due to Heinrich Hertz in his paper, “On the contact of elastic solids” in 1882 [2]. At that
time Hertz was studying Newton’s optical interference fringes in the gap between two
glass lenses. Primarily, he was concerned at the possible influence of elastic deformation
of the surfaces of the lenses due to the contact pressure between them. His theory, which
was worked out during the Christmas vacation in 1880 aroused considerable interest
when it was first published. Hertz made certain assumptions in his theory that are:
•
The surfaces are continuous and non-conforming: a<< R;
•
The strains are small: a<<R;
•
The bodies are in frictionless contact: qx = qy= 0;
•
Each solid can be considered as an elastic half-space: a<<R1,2, <<1;
•
The displacements and stresses must satisfy the differential equations of
equilibrium for elastic bodies, and the stresses are localized.
10
11
Figure 2-1 Two bodies in Hertzian Contact Source: Modified from K.L. Johnson [5]
Here a is the contact area, R is the relative radius of curvature, R1 and R2 are
significant radii of each body and l is significant dimensions of the bodies both laterally
and in depth.
In addition to static loading Hertz also investigated the quasi-static impacts of
spheres that is mentioned in his paper, “On the contact of rigid elastic solids and on
hardness” [1]. In this paper, Hertz also attempted to use his theory to give a precise
definition of hardness of a solid in terms of the contact pressure to initiate plastic yield in
the solid by pressing a harder body into contact with it. This definition has proved
unsatisfactory, as it is very difficult to detect the point of first yield under the action of
contact stress.
Consider the deformation as a normal load P is applied as shown in Fig 2.1 [5].
Two solids are chosen that are of general shape and convex for convenience. Before
12
deformation the separation between two corresponding surface points S1(x,y,z1) and
S2(x,y,z2) is given by equation:
h
2
A ⋅ x + B⋅ y
2
1
2⋅ R'
1
2
⋅x +
R''
⋅y
2
(2.1)
where A and B are positive constants and R' and R'' are defined as principle
relative radii of curvature.
From the symmetry of this expression about the center, the contact region must
extend an equal distance on either side of the center. During the compression distant
points in the two bodies T1 and T2 move towards the center, parallel to the z-axis, by
displacements δ1 and δ2 respectively. If the solids did not deform their profiles would
overlap as shown by dotted lines in Fig 2.1. Due to contact pressure the surface of each
body is displaced parallel to z by an amount uz1 and uz2 (measured positive into each
body) relative to the distant points T1 and T2. If, after deformation, the points S1 and S2
are coincident within the contact surface then,
u z1 + u z2 + h
δ1 + δ2
(2.2)
Writing δ = δ1+δ2 and making use of Eq. (2.1) we obtain an expression for elastic
displacements:
u z1 + u z2
2
δ − A ⋅ x − B⋅ y
2
(2.3)
where x and y are common coordinates of S1 and S2 projected onto the x-y plane.
If S1 and S2 lie outside the contact area so that they do not touch then,
2
u z1 + u z2 > δ − A ⋅ x − B⋅ y
2
(2.4)
Hertz formulated the conditions expressed by equations (2.3) and (2.4) which
must be satisfied by the normal displacements on the surface of the solids. He first made
13
the hypothesis that the contact area is elliptical. Later he introduced that for calculating
the local deformations, each body can be regarded as an elastic-half space loaded over a
small elliptical region of its plane surface. By this simplification, generally followed in
contact stress theory, the highly concentrated contact stresses are treated separately from
the general distribution of stress in two bodies that arises from their shape and the way in
which they are supported. It also gives the well-developed methods to solve boundaryvalue problems for the elastic half-space for the solution of contact problems. In order for
this simplification to be justifiable two conditions must be satisfied:
The significant dimensions of the contact area must be small compared
with the dimensions of each body
with the relative radii of curvature of the surfaces.
A little caution should be taken while applying the results of the theory to low
modulus materials like rubber where it is easy to produce deformations that exceed the
restriction to small strains. Finally, the surfaces are assumed to be frictionless so that only
a normal pressure is transmitted between them.
Thus Hertz can be called as the ‘father of contact mechanics’ whose theory has
proved very important to the mathematicians, scientists, and physicists for future
research. They used Hertz theory in the second half of the 20th century and removed its
restrictions.
Hertz generalized his analysis by attributing a quadratic equation to represent the
profile of two opposing surfaces and gave particular attention to the case of contacting
spheres. For this case, the required distribution of normal pressure σz is:
14
σz
−3
pm
2
2
⋅ 1−
r
a
2
,r ≤ a
(2.5)
The distribution of pressure reaches a maximum (1.5 times the mean contact
pressure pm) at the center of the contact and falls to zero at the edge of the circle of
contact (r = a). Hertz did not calculate the magnitudes of the stresses at points throughout
the interior but offered a suggestion as to their character by interpolating between those
he calculated on the surface and along the axis of symmetry.
In 1953 the book by L.A. Galin [3] appeared in Russian summarizing the
pioneering work of Muskhelishvili in elastic contact mechanics. An up-to-date and
thorough treatment of the same field by Gladwell [4] was published in 1980. These books
exclude rolling contacts and are restricted to perfectly elastic solids.
A Review of Articles on Plane Contact Problems
The plane contact problems are of the following types [3]:
•
Contact problems taking no account of forces of friction,
•
Contact problems for the case when the punches are rigidly connected to
the elastic body,
•
Contact problems of the theory of elasticity considering friction and
•
Problems of contact between two elastic bodies.
The detailed reference study of plane contact problems done here has been written
by dividing them into the above four categories.
Contact Problems Neglecting Forces of Friction
The solution of contact problems is simple if we neglect the forces of friction
between the contacts [3]. This neglect is justified in most of the cases of contact problems
that arise in calculating details of machines. There is a layer of lubricant between
15
machine parts in contact that lowers the force of friction drastically. If the velocity of the
movement of one part is very small as compared to the other, the hydrodynamic
phenomenon taking place in this layer can be neglected. The presence of lubricant
actually means that the forces of friction between the bodies are very small. Therefore it
is possible to equate them to zero with sufficient degree of accuracy and that was one of
the assumptions made by Hertz.
M.A. Sadovsky [9] has studied several cases of a rigid body, exerting pressure on
an elastic semi-infinite plane in the frictionless case. He considered the pressure of the
punch with a plane base on an elastic semi-infinite plane, and also the case when there
are infinitely many punches and they repeat themselves periodically. Actually,
S.A.Chaplygin’s [10] second edition of the collected works contains the solution of the
pressure of the punch with a plane base but he never published his work. The manuscript
was dated 1900, i.e. much earlier than the work by M.A. Sadovsky, which appeared in
1928.
A.I. Begiashvili [11] in 1940 generalized the conclusions obtained in the course of
solving the contact problems for one punch to include the case where the number of
punches is any number desired. He cited examples of punches with a plane base.
All problems, described above were applied to an isotropic semi-infinite plane.
Contact problems for an anisotropic semi-infinite plane neglecting friction were
discussed by G.N. Savin [12] in 1939. He gave the solution of problems about the
pressure of a punch with a plane base, and also with a base bounded by the arc of a circle,
on an anisotropic semi-infinite plane.
16
If the punch moves with a constant speed along the boundary of an isotropic
elastic semi-infinite plane, then the mixed problem, which has to be solved, turns out to
be close to the one which has to be dealt with as in the case of an anisotropic semiinfinite plane. L.A. Galin considered this problem in 1943 and gave the solution for it in
[3]. Here, if friction is absent between the punch and the elastic body, the distribution of
pressure turns out to be the same as in the case of an immobile punch.
In 1976, the report by US DOT [13] presented a general numerical method of
solution to conformal frictionless contact problems. In particular, the method developed
was to be used in future research on the analysis of interfacial contact stresses between a
railway wheel and rail. Numerical influence functions needed for the solutions of
problems with cylindrical and spherical surface geometries were generated and their
accuracy was verified by comparison to exact analytical solutions when they existed.
Also, it was shown that the method in which sphere indents a spherical seat and a
cylinder indents a cylindrical seat produces accurate values of contact pressure approach,
displacements, strains, and applied force. Moreover, a problem of a pitted sphere
indenting a sphere was solved for the first time and the appropriate boundary iteration for
multiply connected contact region was established.
Contact Problems for the Case When the Punches are Rigidly Connected to the Elastic
Body
In this category of problems, it is assumed that if the coefficient of friction
between the bodies in contact is very large, then the bodies turn out to be rigidly
connected to each other. In some cases this type of boundary conditions can approximate
those, which occur between foundations and ground. But in most of these cases, friction
is neglected.
17
A problem of the type for an elastic semi-infinite plane, when displacements on
one part of the boundary and stresses on another part are given, was first considered by
N.I. Muskhelishvili [14] in 1935. It was reduced to the solution of an infinite system of
simultaneous linear equations.
Effective solutions have been developed for the case when the region occupied by
the elastic body is bounded by a circumference. Here, the work of I.N. Kartzivadze
[15,16] in 1943, who gave the solution of the problem for the inside of the circle, and the
work of B.L. Mintsberg [17] in 1948, who gave the solution for the outside of the circle
should be mentioned.
Solutions of mixed problems for an anisotropic semi-infinite plane in the case
when the displacements are given on one part of the boundary and stresses on another
part are given by L.A. Galin [3].
Contact Problems in the Theory of Elasticity Considering Friction
Friction often occurs when the elastic bodies touch each other. Here two cases can
arise. In the first case, one elastic body moves with respect to the other, and the
movement is so slow that the dynamic effect can be neglected. In this case, it can be
supposed that under the action of the displacing force, the punch is situated on the surface
of the elastic body in a state of limited equilibrium. In the second case, no displacement
of the punch as a whole with respect to the elastic body takes place. However at the
points where the tangential effort is less than the normal pressure multiplied by the
coefficient of friction, rigid linkage takes place, i.e. stick occurs and the points where the
tangential effort is sufficient to achieve motion, a displacement of the elastic body with
respect to the punch takes place i.e. slip occurs.
18
The contact problem when forces of friction obeying Coulomb’s law act over the
whole area of contact has been solved by N.I. Muskhelishvili [18] in 1942. This problem
as well as the contact problems discussed earlier can be reduced to finding one function
of complex variable.
When the punch moves along the boundary of the elastic body, the force of
friction acts in a constant direction over the whole area of contact. Consider the problem
of a punch moving with constant speed along the boundary of the elastic semi-infinite
plane. The solution of this problem allows us to establish the extent to which it is
important to consider the dynamic phenomena taking place here. L.A. Galin solved this
kind of problem and the problem with forces for friction for an anisotropic semi-infinite
plane in 1943 and the results are discussed in his book [3].
If the punch is pressed into the elastic semi-infinite plane and there occurs the
forces of friction obeying Coulomb’s law, then area of contact is divided into several
sectors. At places where tangential forces are inadequate for shifting particles of the
elastic body with respect to the punch, these bodies are joined by means of a rigid linkage
and they stick. In this case,
T( ξ)
N( ξ)
<µ
(2.6)
where N(ξ) and T(ξ) are the normal and tangential components of the stress and µ
is the coefficient of friction.
T( ξ)
N( ξ)
≥µ
,
(2.7)
19
Then sliding movement of the punch with respect to the elastic body will occur
and the direction of this sliding will be different at different sectors. One problem of this
type with a plane base has been solved by L.A. Galin in 1945. The solution is given in
[3]. Somewhat different assumptions are made in the work by S.V. Falkovich [19] in
1946. He mentioned that in some sectors friction is absent and so those are the regions of
slip while in others it is present and so those are the regions of stick. However, in case of
machine constructions, a layer of lubricant separates the bodies in contact. Thus in such a
case it is necessary to solve a problem of a theory of elasticity as a problem in the
hydrodynamic theory of lubricants.
Seireg [20] treats friction, lubrication, and wear as empirical phenomena and
relies heavily on the experimental studies done by him to develop practical tools for
design. He summarizes the relevant relationships necessary for the analysis of contact
mechanics in smooth and rough surfaces, as well as the evaluation of the distribution of
the frictional resistance over the contacting surfaces due to the application of tangential
loads and twisting moments. He also presents an overview of the mechanism of the
transfer of frictional heat between rubbing surfaces and gives equations for estimating the
heat partition and the maximum temperature in the contact zone. The problem of friction
and lubrication in rolling/sliding contacts is also given by Seireg [20] where he gives
empirical equations for calculating the coefficient of friction from the condition of pure
rolling to high slide-to-roll ratios. The effect of surface layers is taken into consideration
in the analysis.
Seireg and Weiter [21] conducted experiments to investigate the loaddisplacement and displacement-time characteristics of friction contacts of a ball between
20
two parallel flats under low rates of tangential load application. The tests showed that the
frictional joint exhibited ‘creep’ behavior at room temperatures under loads below the
gross slip values, which could be described by a Boltzmann model of viscoelasticity.
They also found that the static coefficient of friction in Hertzian contacts was
independent of the area of contact, the magnitude of the normal force, the frequency of
the oscillatory tangential load, or the ratio of the static and oscillatory components of the
tangential force [22,23]. Walter Sextro’s book [24] considers the dynamical contact
problems with friction. Here the dynamics of the elastic bodies in contact are described
by a reduced order model through the so-called modal description. Dynamics of
oscillators with elastic contact and friction and rolling contact is also considered.
Abdallah A. Elsharkawy [25] presented a numerical scheme based on Fourier
transformation approach to investigate the effect of friction on subsurface stresses arising
from the two-dimensional sliding contact of two multilayered elastic solids. The analysis
incorporated bonded and unbonded interface boundary conditions between the coating
layers. Two line contact problems were presented. The first one was the contact problem
between a rigid cylinder and a two-layer half space and the second one is the indentation
of a multilayered elastic half space by a rigid flat punch. He presented the effect of
surface coating on the contact pressure distribution and subsurface stress fields.
Problems of Contact Between Two Elastic Bodies
In the works that are mentioned in the previous sections, it was supposed that one
of the bodies is absolutely rigid. It is called as a punch. This assumption imposes certain
limitations, as in reality each of the bodies in contact is elastic. Thus the problems of
contact when both the bodies are elastic have to be considered.
21
The problem of the contact of elastic bodies was first considered by Hertz in 1882
and it has already been discussed earlier in this section in detail. Here, the solution of the
plane problem where two parabolic cylinders whose axes are parallel touch each other
can be found. Later the problem of the contact of elastic bodies was considered by A.N.
Dinnik [26] in 1906.
I.J. Sjtaerman [27] considered the problem of the contact of two cylinders whose
radii are nearly equal in 1940. Here one of the bodies is the outside and the other is the
inside of the cylinder. He considered the problem twice.
A general method of solution of non-Hertzian, non-conformal elastic contact
problems was developed by Singh and Paul [28] in 1974. They considered the classical
contact criterion (which includes that of Hertz) for arbitrary surface geometries. In order
to solve the governing integral equation of the first kind they introduced three different
numerical schemes. The first “simply-discretized method” was found to be relatively
unstable for the particular problems they investigated. In order to overcome this
difficulty, Singh and Paul [29] in 1973-74 introduced two other techniques of solving ill
posed integral equations, called “Redundant Field Point method” and the “Functional
Regularization method”; the latter of which is based on Tychonov’s regularization
procedure.
In the group of conformal problems an elastic sphere indenting an elastic seat has
been solved by Goodman and Keer [30] in 1965. They presented the results for the half
angles of contact up to 20 degrees and provided experimental results that generally agree
with their solutions. It is noted that there are higher order terms in the exact formulation
of the sphere problem that do not appear in the formulation if the half space assumption is
22
used without truncating terms. These terms are particular to the spherical geometry.
Goodman and Keer justify their extension of the Hertzian theory through analysis of
these second order terms.
With the advent of the digital computer several numerical techniques have been
developed to analyze a more general class of contact problems. Conry and Seireg [31] in
1971 have examined elastic contact in terms of a linear programming model. Their
method is general in scope; however, the only examples that were analyzed were
Hertzian or one-dimensional beam problems. Kalker and Van Randen [32] in 1972
derived a variational principle for both linear and non-linear elastic contact problems. For
the case of linear elasticity the principle takes the form of an infinite dimensional convex
quadratic programming problem. They successfully solved both Hertzian and nonHertzian problem. It was concluded that the solution yielded accurate values of approach,
maximum pressure and applied force; however, the actual contact area was not
determined with great accuracy.
A Review of Articles on Three-Dimensional Contact Problems
The solutions of the three-dimensional problem are comparatively difficult to
solve as compared to the plane contact problems. They have characteristics of all spatial
problems of the theory of elasticity. In such problems it is assumed that one of the elastic
bodies occupies a semi-infinite space and the second body in contact, which maybe
absolutely rigid or elastic, can also be replaced by a semi-infinite space. Usually friction
is supposed to be absent between the elastic bodies. It was possible only for certain class
of problems to find a solution of a space axis-symmetrical contact problem for the cases
when friction is considered.
23
If it is assumed that there is no force of friction between the contacting bodies
then the problem indicated is reduced to a certain space problem in the theory of
potential. Here it is necessary to find harmonic function, which tends to zero at infinity
and which has a value on both sides of a plane region of the outline of the area of contact.
Most of the results of the three-dimensional contact problems were obtained by
Russian scientists. A number of problems were investigated by A.N. Dinnik [26]
I.J. Sjtaerman [33] in 1939 gave the solution of the contact problem, in which one
of the contacting bodies is a paraboloid whose degree is greater than the second body.
Hertz in his theory gave the assumption that the radii of curvature of the surfaces
bounding the touching bodies are large in comparison with the area of contact. Therefore
only the first terms were preserved in the equations of these surfaces, and the problem
was reduced to the contact of two bodies bounded by the surfaces of the second order like
elliptical paraboloids or elliptical cylinders. But these considerations are useful for
comparatively smooth bodies. In the case of less smooth bodies; the unevenness on the
surfaces might be of the same order as the dimensions of the area of contact. Therefore
the initial assumption about the approximation of the equation of the surface does not
hold. Moreover, area of contact can be very large in some problems. Thus, a more exact
expression for the shape of the surface of the touching bodies should be taken.
The generalization of the problem considering the pressure of a punch having a
circular cross-section on an elastic semi-infinite body is given in L.A. Galin in [3]. Here
the case is considered where the base of the punch is bounded by a surface that is
represented by a function of two coordinates. The case in which the punch exerting the
pressure on an elastic body is a solid of rotation was investigated in detail by L. A. Galin.
24
In addition, the axi-symmetrical problem with the forces of friction for such punches has
been considered. In this case, the punch, which is pressed against the elastic body, rotates
about its own axis, while the forces of friction arising possessing axial symmetry. Also,
the influence of a load (acting outside of the punch) on the distribution of pressure arising
under the punch, as applied to punches of circular cross-section, has been considered.
Punches of elliptical cross-section have also been investigated. Here it was shown that if
the equation of the base of a punch is a certain polynomial, then the distribution of
pressure under the punch could also be expressed in the form of a polynomial of the same
degree, multiplied by a simple algebraic function (1947). Also the solution to the problem
of the pressure on an elastic semi-infinite body of a wedge-shaped punch has been given
by L.A.Galin [3]. This solution permits the establishment of a law of distribution of
pressure in the vicinity of a vertex point of a punch of a polygonal cross-section
The case where the punch has a narrow cross-section has also been investigated
by L.A. Galin [3]. An example of such a problem is when a narrow beam exerts pressure
on a semi-infinite elastic body. In all the problems indicated earlier it was assumed that
the elastic body on which the punch exerts pressure is sufficiently large to be represented
by a semi-infinite body. Another case is when a rigid body presses on a thin lamina that
was investigated by L.A Galin again in 1948.
L.A. Galin’s work [3] considers a thorough discussion of the problems in the field
of elastic contact mechanics that summarized the work of Muskhelishvili. Even the work
by Gladwell [4] involves a detailed of discussion of it. But these books are restricted to
perfectly elastic solids only.
25
Yen-Yih Ni [34] in his dissertation utilized the mathematical programming
approach for evaluating the Hertzian elliptical contact parameter for conditions with very
large ratio between the major and minor axes. The problem of transition from elliptical to
rectangular contact for the case of crowned cylinders was investigated. The pressure
distribution between short cylinders on semi-infinite solids and between the layers of leaf
springs was also considered. The study basically dealt with developing a computational
procedure for the analysis of pressure distribution between elastic beams in contact as
well as the surface modifications necessary to generate uniform pressure between them.
The contact between elastic solids with finite dimensions is also considered, and a scaling
relationship is developed to improve the computational accuracy when a limited number
of nodes are used as well as when the elements in the grid have large aspect ratios.
A methodology for surface and sub-surface stress calculation of nominally flat on
flat rough surface contact was developed by Maria M. -H. Yu and Bharat Bhushan [35].
This methodology is applicable for both Hertzian contact (large area contact) and point
load contact (small area of asperity contact) with and without surface friction. A threedimensional numerical model was presented by Wei Peng and Bharat Bhushan [36] to
investigate the contact behavior of layered elastic/plastic solids with rough surfaces. The
surface and subsurface stresses in the layer and the substrate were determined and von
Mises yield criterion was used to determine the onset of yield. The surface deformation
and pressure distributions were obtained based on a variational principle with a fast
Fourier transform (FFT)-based technique.
Examples of Numerical Approaches to Contact Mechanics Problems
Finite element techniques have also been developed to solve contact problems by
Chan and Tuba [37] and by Chaud, Haug and Rim [38]. Both methods are general as they
26
handle problems that fall into the domain of FE analysis such as analyzing non-isotropic,
non-homogeneous media or problems with plasticity and creep, however, both works
report only examples which are composed of isotropic materials stressed within the range
of linear elasticity. Chan and Tuba compared their computed results to photo-elastic
studies and concluded that trends were identical but the results lacked close agreement.
Chaud et al, analyzed the non-Hertzian problem of a human knee joint and the contact
between two half spaces where one half space has three bumps on the surface. The
contact area in the latter case found in photo elastic studies had good general agreement
with their computed results.
In the case of conformal contacts a number of problems involving a disc in an
infinite plate under tension have been solved by FEM. Chan and Tuba [37] again
analyzed a plate under tension with a shrink fit disc located in the center. They presented
results that showed good agreement between their computed values of circumferential
stress and the exact solution, however, there is a larger discrepancy between the
computed value of compressive stress and the exact solution. Chaud et al in 1974 have
analyzed the problem of a plate under tension with either a loose or full inclusion. They
show good agreement between their predicted contact stress and experimental results for
a contact angle of 20 degrees.
Sen, Aksakal and Ozel [39] solved an elastic-plastic problem in elastic-work
hardening layered half-space indented by an elastic sphere using FEM. The case of a
surface layer stiffer than the substrate was considered and general solutions for the
subsurface stresses and deformation fields were presented for a relatively thin elastic
27
layer. Differences between the elastic and elastic-plastic solutions for the contact pressure
distribution have been investigated for various layer thickness.
K.L. Johnson’s book [5] covers the theoretical development on normal contact of
elastic solids (Hertzian Contacts) as well as non-Hertzian Contacts. It also includes the
details of dynamic effects and impact, thermo elastic contact, and study of rough
surfaces. Thus, K.L. Johnson’s [5] book covers a detailed and comprehensive study of the
field of contact mechanics.
An Overview of Lekhnitskii’s Work
S.G. Lekhnitskii’s work [6] is discussed separately as this study adapts his
analytical solution method to solve anisotropic contact problems. One of his basic
contributions is his extension of N.I. Muskhelishvili’s work in the plane theory of
isotropic elasticity to the anisotropic case. Here, the author utilizes the theory of analytic
functions of several complex variables in an elementary and systematic manner in order
to solve some boundary value problems. It is of interest to point out that the theory of
functions of several complex variables that ordinarily is considered to be in the domain of
pure mathematics now finds application to problems in modern physics as well as to
problems of technology including contacting mechanics.
Lekhnitskii in his book [6] gives a detailed account of certain parts of the theory
of anisotropic bodies, which have been studied, but not organized systematically. The
problems of plane deformation and generalized plane stress are discussed briefly. In all
cases it is assumed that the deformations are elastic and small, and the material satisfies a
generalized Hooke’s law. The analytical solution described in Lekhnitskii [6] is adapted
for this study and explained in detail in the next chapter.
CHAPTER 3
ANISOTROPIC ELASTICITY EQUATIONS FOR FCC SINGLE CRYSTALS
Analytical solutions have the advantage of providing a closed-form solution for
some contact problems. However, very complex problems often do not have exact
analytical solutions; many approaches represent a combination of theoretical and
empirical solutions, or close approximations. For the problem considered in this study, a
semi-analytical solution procedure with complex functions is outlined for which closed
form solution is not possible. The complex parameters are the quantities, which depend
on the elastic constants. These functions determine the directions and deformations in a
homogeneous body bounded by a cylindrical surface. The problems, which consider the
equilibrium of such a body, are reduced to determination of functions that satisfy
differential equations of the second, fourth, or sixth orders of the type:
a0
∂ 2mu
∂ 2mu
∂ 2m u
a
......
= f ( x, y )
+
+
a
+
1
2m
∂y 2 m
∂x 2 m −1∂y
∂x 2 m
(3.1)
Here m=1, 2, or 3; a0, a1,….a2m are constant coefficients which depend on the
elastic constants; u is an unknown stress function; f (x, y) is a given function of the
coordinates. The roots µ k , µ k that are always complex or purely imaginary are called the
complex parameters of the equation:
a 2 m µ 2 m + a 2 m −1 µ 2 m −1 + ..... + a0 = 0
(3.2)
28
29
Slip Activation and Deformation in FCC Single Crystal
In a single crystal material, with a FCC crystal structure, the material exhibits
cubic symmetry with a particular kind of anisotropy. The slip activation depends on the
loading orientation and the stress-strain pattern in a material depends on the activated slip
systems. Thus stress-strain behavior is a function of load orientation and varies with it.
(Fig 3.1). In an isotropic material the twelve primary slip planes (or fewer depending on
orientation), should be activated simultaneously based on equal Schmid factors (Fig 3.2
and Fig 3.3). The Schmid factor, µ, is a function of the load orientation, the slip plane
orientation and the slip direction:
µ = cosλ.cosφ
(3.3)
τ rss = m.σ
(3.4)
where σ is the applied load, τ rss is the RSS (resolved shear stress) component in a
given slip plane and direction, λ is the angle between the direction of the applied load and
the shear direction, and φ is the angle between the applied load and the normal to the slip
plane (Fig 3.2).
Figure 3-1 Orientation-dependant stress-strain behavior Source: Modified from Shannon
Magnan’s Masters thesis [40], 2002
30
Figure 3-2 Load and slip directions and angles Source: Modified from Shannon
Magnan’s Masters thesis [40], 2002
Figure 3-3 Primary resolved shear stress planes and directions Source: Modified from
Stouffer and Dame [41], 1996
31
The cosine of the angle between two directions [h1 k1 l1] and [h2 k2 l2] can be
found using the direction indices [40]:
cos θ
h 1⋅ h 2 + k 1⋅ k 2 + l 1⋅ l 2
2
2
2
2
2
2
h 1 + k1 + l1 ⋅ h 2 + k2 + l2
(3.5)
Since the CRSS is reached when the RSS is equal to the yield stress of the
material, the slip systems with the highest Schmid factors will reach the CRSS first. This
is called as “Schmid’s Law” (Eq 3.5). But this law is true only for isotropic materials, as
its correlation to non-isotropic materials is not yet known.
Elasticity
Elasticity of anisotropic FCC crystal varies with orientation. In any given
direction the spacing between atoms in an FCC crystal is different. For example, consider
Fig 3.4. A is the center of the front face; B of the right face and C is the center of the FCC
cube. If ao is the side of the cube or the unit atomic spacing then we have the distance
between A to B as ao/√2 and the distance from A to C as ao/2. Another way of expressing
the relative spacing between atoms is given in the Table 3.1.
Figure 3-4 FCC crystal structure Source: Modified from Shannon Magnan’s Master
thesis [40], 2002
32
Table 3-1 Atomic densities on FCC crystal planes
FCC Plane
Atom/Area
Atom/Area
{100}
2/ao2
2
{110}
2/(√2.ao2)
1.414
{111}
4/(√3.ao2)
2.309
Source: Dieter [42], 1986
Planes like {111} are called “closed-packed planes,” because they minimize the
spacing between atoms; they are the most common slip planes because the atoms do not
have to travel great distances to reach another atomic position. Closed-packed directions,
like closed-packed planes, minimize the distance between atoms. Therefore slip often
occurs along the close-packed directions in close-packed planes to minimize the amount
of energy needed for displacement.
Elasticity Matrix
The energy needed to deform in any direction is related to the elastic constants,
and any material can be completely defined with 36 separate elastic constants. However,
most material structures obey some type of symmetry, which reduces the number of
independent constants. Dieter [42] has given the rotational symmetry of various crystal
structures (Table 3.2).
33
Table 3-2 Symmetry in various crystal structures.
Rotational
Crystal structure
symmetry
No. of Independent
Elastic Constants
Tetragonal 1 fourfold rotation
6
Hexagonal 1 sixfold rotation
5
Cubic
3
4 threefold rotation
Isotropic
2
Source: Dieter [42], 1986
As indicated in Table 3.2 isotropic materials have only two independent elastic
constants (E and G, E and ν, or ν and G for instance). Cubic structures require three
independant elastic constants (E, ν and G).
According to Lekhnitskii [6] when no elements of elastic symmetry are present in
the general case of a homogeneous anisotropic body, the original elasticity matrix [aij] is
given by,
a 11
a 21
a
31
a ij
a 41
a 51
a
61
a 12 a 13 a 14 a 15 a 16
a 22 a 23 a 24 a 25 a 26
a 32 a 33 a 34 a 35 a 36
a 42 a 43 a 44 a 45 a 46
a 52 a 53 a 54 a 55 a 56
a 62 a 63 a 64 a 65 a 66
(3.6)
The subscripts i and j correlate to the stress and strain components. The strain is
related to stress by,
εi
a ij⋅ σ j
(3.7)
34
The constants aij are called as the coefficients of deformation. If an elastic
potential exists, the number of elastic constants in the most general case of anisotropy is
reduced to 36. Such an elastic potential exists when the variation of the body under
deformation occurs isothermally or adiabatically.
Considering equilibrium, if we assume isothermal deformation, i.e. the
temperature of each element remains constant, we have,
a ij
a ji
(3.8)
Therefore the matrix is reduced to 21 components:
aij
a11
a12
a
13
a14
a15
a16
a12 a13 a14 a15 a16
a22 a23 a24 a25 a26
a23 a33 a34 a35 a36
a24 a34 a44 a45 a46
a25 a35 a45 a55 a56
a26 a36 a46 a56 a66
(3.9)
This is the general elasticity matrix for an anisotropic material under isothermal
deformation.
For FCC crystals exhibiting cubic syngony, the number of independent constants
reduces to three for the final elasticity matrix [6]:
a 11
a 12
a
12
a ij
0
0
0
a 12 a 12
0
0
0
a 11 a 12
0
0
0
0
0
a 44
a 12 a 11 0
0
0
0 a 44 0
0
0
0
0
0 a 44 0
0
0
(3.10)
The constants are defined by E, the modulus of elasticity, G, the shear modulus
and ν, Poisson’s ratio along the given directions:
35
a 11
a 44
a 12
1
E xx
(3.11)
1
G yz
−
ν yx
E xx
(3.12)
−
νxy
E yy
(3.13)
The main aim of this study is to find the state of subsurface stresses in the
material coordinate system of single crystal material, and then using those stresses to
calculate the resolved shear stresses (RSS) on the twelve primary slip systems. For an
isotropic material a single elastic constant governs the transformation from stress to strain
as given in Eq. 3.7. However, for an anisotropic material, the analysis of stress and strain
fields is not so straightforward. First of all the material properties in a desired orientation
is determined. If the material and the specimen coordinate axes are in the same directions
then the problem is easy, as the transformations are not needed. The stress-strain relation
for an anisotropic solid with cubic symmetry has three independent constants in the
material coordinate system, as well as a “stress tensor matrix,” instead of a single
elasticity constant as in the case of an isotropic material. Thus, if the specimen orientation
varies, then it is necessary to convert between the specimen and material coordinate
systems.
Coordinate System Transformation
The first step in defining the elasticity matrix is to determine the precise
orientation of the specimen or part, either in terms of the material Miller indices
(direction indices) or angular measurements. Physically it is nearly impossible to cut a
sample such that the x, y, and z test axes are perfectly aligned to the material axes: [100],
36
[010] and [001] respectively. Thus it is necessary to transform the known specimen
stresses to the material coordinate system. The stresses vary by slip plane and direction
according to a particular plane’s cross-sectional area and its given orientation within a
unit cube of the material. Therefore, these stresses are not affected by anisotropy.
However, only two material properties are independent without anisotropic effects; the
shear coupling induced in the three-dimensional model, and resulting component stresses,
will not be properly accounted for without the third independent constant to define the
single crystal material. Lekhnitskii [6] gives a detail explanation on this topic.
The transformation equations presented here follow the procedures mentioned in
Lekhnitskii [6] (1963) and Stouffer and Dame [41] (1996). The transformation from the
specimen to the material coordinate system can be accomplished by two methods. In the
first one, the angles between the original and transformed coordinate systems may be
directly measured to find the direction cosines. This method is possible only if the angles
can be easily found out. The second approach, based on the rigid body rotations, may be
used for more complex orientations, where the angles between the two coordinate
systems are not as obvious. But here one should know the Miller’s indices of the
transformed axes. In this method, the axes are rotated through a series of steps to arrive at
the final transformed matrix.
Coordinate transformations can be performed as long as the orientation of the
angle is known, and then the final transformed matrices can be used to determine the
stresses and strains resolved on any given plane or slip system. The original coordinate
system is referred as the material coordinate system and the transformed coordinate
system is called as the specimen coordinate system. The specimen coordinate system is
37
offset by some angular displacement from the material coordinate system. The material
axes are denoted by: xo = [100], yo = [010], and zo = [001] while the specimen axes are
denoted by x″, y″ and z″ (Fig 3.5).
x″
Figure 3-5 Material (xo, yo, zo) and specimen (x″, y″, z″) coordinate systems.
Steps in Coordinate Transformation
Shannon Magnan [40] has clearly explained the steps involved in transformation.
The easiest way to do the transformation is to break it into several rigid body rotations.
The process we have used is a three-step process. The first transformation, to the x, y and
z axes, is performed by rotating the material coordinate axes by ψo about the zo-axis
(positive direction is defined as xo towards yo)(Fig 3.6).
+
Figure 3-6 First rotation of the material coordinate axes about the zo-axis.
38
The transformed coordinates, in terms of the original coordinates, are:
( )
( )
x
x o⋅ cos ψ o + y o⋅ sin ψ o
y
−x o⋅ sin ψ o + y o⋅ cos ψ o
(3.15)
z
zo
(3.16)
( )
( )
(3.14)
Writing the first step in matrix form:
x
y
z
cos ( ψ o) sin ( ψ o) 0 x o
−sin ( ψ o) cos ( ψ o) 0 ⋅ y o
1 zo
0
0
(3.17)
The second transformation is obtained by reflecting the load vector to the x-z
plane and rotating the x, y, and z-axes by ψ1 to x′, y′ and z′-axes about the y-axis
(positive direction is defined as z towards x) (Fig 3.7).
Figure 3-7 Second rotation about the y-axis.
Writing the second step in matrix form:
x'
y'
z'
cos ( ψ 1) 0 −sin ( ψ 1) x
1
0
0
⋅ y
sin ( ψ 1) 0 cos ( ψ 1) z
(3.18)
39
The third and the final step occur by rotating the y′ and z′-axes by ψ2 to the x″, y″
and z″ axes about the x′ axis (positive direction is defined as the y′ towards z′) (Fig 3.8).
Figure 3-8 Third rotation about the x′-axis.
The matrix for the step is:
x''
y''
z''
0
0
1
x'
0
cos
ψ
sin
ψ
( 2) ( 2) ⋅ y'
0 −sin ( ψ 2) cos ( ψ 2) z'
(3.19)
The total transformation can then be calculated by multiplying the three
individual step matrices together (Note: the first transformation becomes the last one
multiplied):
x''
y''
z''
α 1 β 1 γ 1 xo
α 2 β 2 γ 2 ⋅ y o
α β γ z
3 3 3 o
(3.20)
where,
α 1 β 1 γ 1
α 2 β 2 γ 2
α β γ
3 3 3
0
0
1
cos ( ψ 1) 0 −sin ( ψ 1) cos ( ψ 0) sin ( ψ 0) 0
1
0
0 cos ( ψ 2) sin ( ψ 2) ⋅ 0
⋅ −sin ( ψ 0) cos ( ψ 0) 0
0 −sin ( ψ 2) cos ( ψ 2) sin ( ψ 1) 0 cos ( ψ 1)
0
1
0
(3.21)
40
The resultant values represent the cosines of the angles between the material and
specimen coordinate system axes (Table 3.3).
Table 3-3 Direction Cosines
xo
yo
zo
x″
α1
β1
γ1
y″
α2
β2
γ2
z″
α3
β3
γ3
Source: Modified from Lekhnitskii [6]
When fewer than three steps are used, the selection of primary and secondary
rotation axes is somewhat arbitrary and it is important to verify the results. Several
checks based on perpendicularity can be calculated to ensure a proper orthogonal
coordinate transformation has been performed (Appendix).
Stress Transformation
Once the direction cosines between the material and specimen coordinate axes are
calculated, the load conditions can be applied and incorporated into separate matrices to
correctly transform the individual stresses and strains. These transformed matrices are
then used to solve for the resolved stresses and strains in each desired slip system.
Lekhnitskii [1] gives the stress transformation as:
{σ″} = [Q′]{σ}
(3.22)
{σ} = [Q′]-1{σ″}= [Q] {σ″}
(3.23)
Here [Q] is the stress transformation matrix, a function of the direction cosines
calculated above:
41
α 2
1
2
β1
2
Q := γ 1
β ⋅γ
1 1
γ 1⋅ α 1
α 1⋅ β 1
2
α2
2
β2
2
γ2
2
α3
2
β3
2
γ3
β 2⋅ γ 2 β 3⋅ γ 3
γ 2⋅ α 2 γ 3⋅ α 3
α 2⋅ β 2 α 3⋅ β 3
2⋅ α 3⋅ α 2
2⋅ α 1⋅ α 3
2⋅ β 3⋅ β 2
2⋅ β 1⋅ β 3
2⋅ γ 3⋅ γ 2
2⋅ γ 1⋅ γ 3
(β 2⋅ γ 3 + β 3⋅ γ 2) (β 1⋅ γ 3 + β 3⋅ γ 1)
(γ 2⋅ α 3 + γ 3⋅ α 2) (γ 1⋅ α 3 + γ 3⋅ α 1)
(α 2⋅ β 3 + α 3⋅ β 2) (α 1⋅ β 3 + α 3⋅ β 1)
2⋅ β 2⋅ β 1
2⋅ γ 2⋅ γ 1
(β 1⋅ γ 2 + β 2⋅ γ 1)
(γ 1⋅ α 2 + γ 2⋅ α 1)
(α 1⋅ β 2 + α 2⋅ β 1)
2⋅ α 2⋅ α 1
(3.24)
The state of stress is defined in terms of the specimen {σ} or material {σ″}
stresses by:
σx
σy
σ
z
σ
τyz
τzx
τ
xy
σ''x
σ''y
σ''
z
σ''
τ''yz
τ''zx
τ''
xy
(3.25)
The strain transformation is carried out in more or less the same manner but we
are not concentrating on strains in this study, so we are not going to discuss the strain
transformation here in detail. Lekhnitskii [6] and Shannon Magnan [40] have given the
detailed explanation of the strain transformation.
Transformation of the Elastic Constants
The elasticity matrix also undergoes transformation but remains symmetric and is
given by:
[cij] = [Q]T [aij] [Q]
A maximum of 21 individual constants may be present, depending on the
specimen orientation. Now this [cij] matrix gives us the component stresses in the
(3.26)
42
specimen coordinate system. The above equations thus can be further utilized to solve for
the component stresses in the material coordinate system.
Slip System Shear Stresses
The component stresses define the complete state of stress for the material, but
these stresses alone do not reveal much about individual slip systems or RSS. The twelve
primary slip planes are defined by both a slip plane and direction.
Resolved Shear Stresses
The resolved shear stresses on twelve primary slip planes {(111)[110]} can be
calculated from the following:
τ 1
τ2
1 0 −1 1
0 −1 1 −1
τ3
1 −1 0 0
τ4
−1 0 1 1
τ5
−1 1 0 0
τ
6 := 1 ⋅ 0 1 −1 −1
τ7
6 1 −1 0 0
0 1 −1 −1
τ8
1 0 −1 −1
τ
9
0 −1 1 −1
τ 10
−1 0 1 −1
−1 1 0 0
τ
11
τ 12
0 −1
1
0
1 −1
σx
σ
−1 y
0 σz
⋅
−1 τ xy
0 τ
zx
−1
τ yz
0
−1
0 −1
−1
−1
−1
1
0
−1
0
(3.27)
1 −1
The stresses resolved onto the primary slip systems are known and can be used to
predict slip behavior within a particular system. The analyses of some crystal orientations
are discussed in detail in the chapter 6
.
CHAPTER 4
GENERALIZED PLANE STRAIN EQUATIONS FOR A HOMOGENEOUS
ANISOTROPIC ELASTIC SOLID
Plane Strain Conditions for Isotropic Contacts
The assumption of the plane strain condition is justified when the thickness of the
solid is large compared with the width ‘2a’ of the contact region. Many books such as
“Contact Mechanics” by K.L. Johnson [5] explains these categories of problems in detail.
For practical problems contact width is very small as compared to the length of the
isotropic body and hence the plane strain conditions are valid. The loading profile and the
shape of the contact region are calculated under these plane strain conditions [5]. The
subsurface stresses can also be evaluated under these conditions, the formulas for which
are given [5].
Under the conditions of plane strain, the equations of equilibrium are given as:
∂σx ∂τxy
+
=0
∂x
∂y
(4.1a)
∂σy ∂τxy
+
=0
∂y
∂x
For compatibility, the corresponding strains εx, εy, and γxy must satisfy the
equation
2
2
∂ 2 ε x ∂ ε y ∂ γ xy
+
=
∂x∂y
∂y 2
∂x 2
(4.1b)
43
44
Figure 4-1 Isotropic elastic half-space
The boundary value problem for evaluating subsurface stresses under the action
of traction forces N(ξ) and T(ξ), will involve solving equations subjected to the
conditions:
σ y = τ xy = 0, x < −a , x > + a
σ y = − N (ξ ) τ xy = −T (ξ ) ,
(4.2)
− a ≤ x ≤ +a
The formulas for calculating subsurface stresses for any pressure profile is given
as [2]:
σx
⌠
−2⋅ y
⋅
π
⌡
∞
−∞
σy
3
−2⋅ y ⌠
⋅
π
⌡
⌠
N( ξ) ⋅ ( x − s )
2
ds − ⋅
2
π
2
2
( x − s ) + y
⌡
∞
−∞
2
∞
T( ξ) ⋅ ( x − s )
( x − s ) + y
2
−∞
2
2⋅ y ⌠
ds −
⋅
2
π
( x − s ) 2 + y2
⌡
N( ξ)
3
∞
−∞
2
2
ds
(4.3a)
T( ξ) ⋅ ( x − s )
( x − s ) 2 + y 2
2
ds
(4.3b)
45
τxy
2 ∞
−2⋅ y ⌠
π
⋅
⌡
−∞
⌠
2⋅ y
ds −
⋅
2
π
( x − s ) 2 + y 2
⌡
N( ξ) ⋅ ( x − s )
∞
−∞
T( ξ) ⋅ ( x − s )
2
( x − s ) 2 + y2
2
ds
(4.3c)
where N(ξ) is the normal loading and T(ξ) is the tangential loading (Fig. 4-1) and
T (ξ ) = µ ∗ N (ξ )
(4.4a)
where ξ is any point on the X-axes
In this study, N (ξ ) = p 0 * 1 −
ξ2
(4.4b)
a2
p0 =
2P
, the maximum pressure
πa
(4.4c)
a=
4 PR
πE *
(4.4d)
where R is composite radius of curvature and E* is composite modulus of elasticity.
Generalized Plane Strain Conditions for Anisotropic Contact Problems
For the case of a cylindrical isotropic body contacting anisotropic elastic halfspace, the conditions of plane strain given by eq.4.1a and eq.4.1b will not be met. The
reason for this is that anisotropic materials can have out of plane stresses induced because
of shear coupling. To account for this effect additional equilibrium equation has to be
included to set of eq.4.1 as shown in eq.4.5 and this describes the condition of
generalized plane strain. It should be noted that even though stresses are not functions of
z there are five independent stresses that can be calculated and σz is a function of these
five stresses. Lekhnitskii has outlined a solution for solving these sets of equations.
46
∂σx ∂τxy ∂τxz
+
+
=0
∂x
∂y
∂z
∂σx ∂τxy
+
=0
∂x
∂y
∂σy ∂τxy ∂τyz
+
+
=0
∂y
∂x
∂z
∂σy ∂τxy
+
=0
∂y
∂x
∂τxz ∂τyz ∂σ z
+
+
=0
∂x
∂y
∂z
∂τxz ∂τyz
+
=0
∂x
∂y
(4.5)
All the stress values except σz are independent in the solution [6] that will be
discussed later in this chapter. The boundary conditions on the surface for this study are
(Fig 4-2):
σy
N( ξ)
τxy
T( ξ)
(4.6)
τxz 0
where N(ξ) is the function for normal loading and T(ξ) is the function for tangential
loading. The set of equations Eq. 4.4 are valid for the anisotropic case also [5].
Figure 4-2 Distribution of forces in an anisotropic half-space
The stress-strain relations for a general anisotropic solid are given as follows [6]:
47
εx
a11⋅ σx + a12⋅ σy + a13⋅ σz + a16⋅ τxy
εy
a12⋅ σx + a22⋅ σy + a23⋅ σz + a26⋅ τxy
εz
a13⋅ σx + a23⋅ σy + a33⋅ σz + a36⋅ τxy
(4.7)
γ yz
a44⋅ τyz + a45⋅ τxz
γ xz a45⋅ τyz + a55⋅ τxz
γ xy
a16⋅ σx + a26⋅ σy + a36⋅ σz + a66⋅ τxy
The above equations include the elastic coefficients aij due to which it is possible
to calculate stresses in different orientations. These aij values are calculated through the
coordinate transformation discussed in the previous chapters and are functions of 3
variables E, G and ν depending on the orientations of the specimen (Eq. 3.9).
Generalized Plane Strain Equations
Consider the elastic equilibrium of a body bounded by a cylindrical surface. The
body is under the influence of body forces and stresses that are distributed along the
surface. The region of the cross section can be either finite or infinite. The body forces
and the surface stresses are assumed to act in planes normal to the generators of the
cylindrical surface and do not vary along the generators. (Fig 4.3)
48
Figure 4-3 Homogeneous Anisotropic Body bounded by a cylindrical surface Source:
Modified from Lekhnitskii [6]
The body is referred to a system of Cartesian coordinates x, y, z which has z-axis
parallel to the generators and the x- and y-axes directed arbitrarily. The components of
the stresses applied to the cylindrical surface are denoted by Xn and Yn per unit area and
the components of the body forces are denoted by X and Y per unit volume. Pz is the
axial force; M1 and M2 are the bending moments while Mt is the twisting moment. Let us
assume that the body forces are derivable from a potential U then,
X
−
Y
−
δU
δx
(4.8)
δU
δy
The system of differential equations, which the stress functions must satisfy and
that has the general solution [1] are given as:
F = F′ + F0
,
ψ = ψ′ +ψ0
(4.9)
Here F′, ψ′ is the general solution of the system of homogeneous equations:
L4F′ + L3ψ′ = 0
,
L3F′ + L2ψ′ = 0
(4.10)
49
and F0 and ψ0 are particular solutions of the non-homogeneous system for which general
solution is given by Eq. (4.9). For detailed analysis please refer Lekhnitskii [6].
The general solution for (4.10) is obtained by solving for those equations
simultaneously. If ψ′ is eliminated we get:
(L4L2 – L32) F′ = 0
(4.11)
The equation for ψ′ is obtained in exactly the same way. The operator of the sixth
order L4L2 – L32 can be decomposed into six linear operators of the first order. Then Eq
(4.11) can be represented by:
D6D5D4D3D2D1F′ = 0
(4.12)
Here,
Dk
δ
δy
− µ k⋅
δ
(4.13)
δx
µk are the roots of the algebraic equation which corresponds to the differential equation
(4.11) and is:
l4( µ ) ⋅ l2( µ ) − l3 ( µ )
2
0
(4.14)
where we have:
2
l2( µ ) := β 55⋅ µ − 2⋅ β 45⋅ µ + β 44
3
(
)
2
(
)
l3( µ ) := β 15⋅ µ − β 14 + β 56 ⋅ µ + β 25 + β 46 ⋅ µ − β 24
4
3
(
)
2
l4( µ ) := β 11⋅ µ − 2⋅ β 16⋅ µ + 2⋅ β 12 + β 66 ⋅ µ − 2⋅ β 26⋅ µ + β 22
Here in Eq. (4.15)
(4.15)
50
β ij
ci3⋅ cj3
cij −
( i, j
(4.16)
c33
1 .. 6)
where βij are the elastic constants or coefficients of deformation for the rotated coordinate
system. If there are no transformations, i.e. the specimen coordinate axes are same as the
material coordinate axes then we can use the following formula also:
β ij
aij −
ai3⋅ aj3
(4.17)
a33
The roots of the system of Eq (4.15) are of the form α + βi where β > 0 and the
root order is selected with the first root having the highest positive or negative value of
α and the second one having the next highest value if there is one. Finally the last root
should have value of α = 0 if there is one.
Lekhnitskii [6] has stated a theorem according to which the roots of the Eq. (4.14)
are always complex and imaginary. Let us assume roots µk as distinct:
D1F′ = ϕ2,
D2ϕ2 = ϕ3,
D3ϕ3 = ϕ4,
D4ϕ4 = ϕ5,
D5ϕ5 = ϕ6
(4.18)
The function ϕ6 satisfies the equation:
D6ϕ6 = 0
(4.19)
The general integral is equal to an arbitrary function of the argument x + µ6 y and is
denoted by fV6(x + µ6 y).
ϕ6 = fV6(x + µ6 y)
(4.20)
D5ϕ5 = fV6(x + µ6 y)
(4.21)
Thus,
By integrating this equation, we get:
51
φ5
f5
IV
(
IV
)
⋅ x + µ 5 ⋅y +
(
)
f 6 ⋅ x + µ 6 ⋅y
(4.22)
µ6−µ5
In the same way we define ϕ4, ϕ3, ϕ2 and finally F′. Changing the notations of the
arbitrary functions, we get the general expressions for F′ and analogous expression for
ψ′:
6
(
∑
F'
)
Fk ⋅ x + µ k ⋅y
k=1
(4.23)
6
(
∑
ψ'
)
ψ k ⋅ x + µ k ⋅y
k=1
The functions F′ and ψ′ satisfy both Eq. (4.23) and Eq. (4.10). Thus:
ψ k ⋅ ( x + µ k ⋅ y)
−
l 3 ( µ k)
l 2 ( µ k)
⋅ F' k ⋅ ( x + µ k ⋅ y) + a k ⋅ ( x + µ k ⋅ y) + b k
(4.24)
The general expressions for the stress functions are now given as:
( ( )
( )
( ) ) + F0
F
2 Re F1 z1 + F2 z2 + F3 z3
ψ
2 Re
λ 1 ⋅F' 1 z1 + λ 2 ⋅F' 2 z2 +
( )
( )
1
λ3
(4.25)
⋅ F' 3 ( z3) + ψ 0
Here Re is the notation for the real part of the complex expression in the brackets, Fk(zk)
are analytic functions of the complex variables zk = x + µk y (k = 1,2,3) and λ1, λ2, λ3 are
the complex numbers equal to:
λ1
−l3 ( µ 1)
l2 ( µ 1)
λ2
−l3 ( µ 2)
l2 ( µ 2)
λ3
−l3 ( µ 3)
l4 ( µ 3)
(4.26)
52
Now let us assume that the body forces are absent. In this case Lekhnitskii [6] has
given formulas that connect the components of stress and displacement with the functions
ϕk where,
Φk ( zk)
F' k ( zk)
Φ3 ( z3)
1
λ3
⋅ F' 3 ( z3)
(4.27)
where (k=1,2)
The component of stresses are given as:
(
2
2
2
)
σx
2 ⋅ Re µ 1 ⋅ Φ' 1 + µ 2 ⋅ Φ' 2 + µ 3 ⋅ λ 3 ⋅ Φ' 3
σy
2 ⋅ Re Φ' 1 + Φ' 2 + λ 3 ⋅ Φ' 3
τ xy
−2 ⋅ Re ( µ 1 ⋅ Φ' 1 + µ 2 ⋅ Φ' 2 + µ 3 ⋅ λ 3 ⋅ Φ' 3)
τ xz
2 ⋅ Re µ 1 ⋅ λ 1 ⋅ Φ' 1 + µ 2 ⋅ λ 2 ⋅ Φ' 2 + µ 3 ⋅ Φ' 3
τ yz
−2 ⋅ Re ( λ 1 ⋅ Φ' 1 + λ 2 ⋅ Φ' 2 + Φ' 3)
σz
(
)
(
−1
c 33
)
(4.28)
⋅ ( c 13⋅ σ x + c 23⋅ σ y + c 34⋅ τ yz + c 35⋅ τ xz + c 36⋅ τ xy)
The Distribution of Stresses in an Elastic Half-Space Under the Influence of
Stresses Applied to the Bounding Plane
Consider the elastic equilibrium of an infinitely homogeneous body bounded by a
plane (an elastic half space). The body is in a state of generalized plane deformation
under the influence of stresses applied to the bounding plane. Considering the x-z plane,
and the y-axis is directed outward (Fig 4.3). Assuming:
The material possesses rectilinear anisotropy of the most general form;
The stresses act on the planes normal to the z-axis and do not vary along this axis;
53
The resultant vector of the stresses distributed in any strip of infinite width
parallel to the z-axis is finite and tends toward a definite limit as the ends of the segment
tend toward infinity.
Figure 4-4 Elastic equilibrium of an infinite homogeneous body with a parabolic profile
Modified from Lekhnitskii [1]
Let f(z) be a function of the complex variable z = x + iy, holomorphic in the lower
half-plane y ≤ 0 and continuous up to the boundary where f(∞) = 0. Then, if z is a point in
the lower half-plane and ξ is a point on the boundary (the abscissa), the following
equalities hold according to Lekhnitskii [6]:
∞
⌠
f (ξ )
⋅
dξ
ξ−z
2 ⋅ πi
⌡− ∞
1
−f ( z)
∞
⌠
f'' ( ξ )
⋅
dξ
ξ−z
2 ⋅ πi
⌡− ∞
1
0
(4.29)
Denote the normal and tangential components of the external stresses (per unit
area) by N(ξ) and T(ξ) respectively. The boundary conditions for y = 0 can be written as
given by Eq. (4.6). Hence after solving we get the final conditions for the functions ϕ′k
and ϕ′′k:
54
Φ' 1 ( z1)
⌠
1
⋅
2 ⋅ π⋅ i⋅ ∆
⌡
∞
N ( ξ ) ⋅ ( µ 3 ⋅ λ 2 ⋅ λ 3 − µ 2) + T ( ξ ) ( λ 2 ⋅ λ 3 − 1)
ξ − z1
dξ
−∞
Φ' 2 ( z2)
⌠
1
⋅
2 ⋅ π⋅ i⋅ ∆
⌡
∞
N ( ξ ) ⋅ ( µ 1 − µ 3 ⋅ λ 1 ⋅ λ 3) + T ( ξ ) ⋅ ( 1 − λ 1 ⋅ λ 3)
ξ − z2
dξ
(4.30)
−∞
∞
Φ' 3 ( z3)
⌠
N ( ξ ) ⋅ ( µ 2 ⋅ λ 1 − µ 1 ⋅ λ 2) + T ( ξ ) ( λ 1 − λ 2)
⋅
dξ
ξ − z3
2 ⋅ π⋅ i⋅ ∆
⌡− ∞
1
Here
∆
(
N( ξ)
zi
)
(
)
µ 2 − µ 1 + λ2⋅ λ3⋅ µ 1 − µ 3 + λ1⋅ λ3⋅ µ 3 − µ 2
2
po⋅ 1 −
x
a
2
,
T( ξ)
2
µ⋅ po⋅ 1 −
x
a
2
(4.31)
x + µ i⋅ y
y = 0 and x = ξ and z, z1, z2, z3 take one and the same value equal to ξ.
Thus the components of stresses can be calculated making use of these systems of
equations (4.30). Further using these stresses and Eq. 3.27 primary RSS can be
calculated.
CHAPTER 5
FINITE ELEMENT SOLUTION OF THE SUBSURFACE STRESSES
Finite Element Model
The finite element method can be used to model specific geometries and
orientations of the bodies in contact. Here, the contact has been simulated by using
parabolic loading and no gap elements have been used in any of the models. Instead,
loading is directly applied which reduces the time for pre-processing and also for solving
the model. Using the ANSYS finite element software (Version 6.1), a few cases have
been solved, for which the results are discussed in the next chapter.
As a stress based process, slip deformation is well explained by the numerical
model’s highest individual resolved shear stresses. The results of a few cases that are
solved analytically using Lekhnitskii’s [6] solution mentioned in the previous chapter is
further used to compare with the ANSYS solution by applying the same loading
conditions in the finite element model.
The assumptions made here are as follows:
•
The contact is sliding
•
The contact width is calculated from the Hertzian solution, which is used
in isotropic case as Lekhnitskii solves for stress solutions and not for
contact problems.
•
The actual contact using gap elements is replaced by the parabolic profile
loading on the body in contact.
55
56
Case 0: δ = 00, γ = 00, θ = 00
Casting Coordinate System
Figure 5-1 Crystallographic Orientations of the anisotropic body
57
Figure 5-2 Global and material coordinate systems
The material properties of anisotropic (single crystal superalloys) vary
significantly with direction relative to the crystal lattice. The anisotropic model for the
crystallographic orientation 1, in which the material and global coordinate axes are the
same and X is [100], Y is [010] and Z is [001] (details explained in chapter 6) is the first
case to be analyzed using FEM to verify the analytical finite element model (Fig. 5-1).
The solution of this anisotropic model is compared with the accuracy of the analytical
solution.
The second crystallographic orientation (crystallographic orientation 4 in chapter
6) is the one in which the body is oriented such that the material and global coordinate
axes are not the same and thus transformations are needed (Fig.5.2). In this case θ = 400
and δ = γ = 10.610 (Fig 5.1). The subsurface stresses are evaluated after proper coordinate
transformations. The results are then compared with the results of the analytical solution.
58
Figure 5-3 Isotropic cylindrical indenter and the anisotropic substrate in contact
The applied load ‘P’ on the cylindrical indenter is 4080lbs for all the cases,
contact width ‘2a’ is 0.02in, maximum pressure ‘po’ is calculated as 2P/πa, and ‘x’ is the
distance of the point on the surface of the body (on the X-axis) from the origin. The
contact model is shown in Fig 5.3. The single crystal turbine blades used in high
temperature applications in aircraft and rocket engines are attached to turbine disks made
of isotropic materials. The contact between the blade and the disk mostly has the width of
0.02in approximately. Hence the considered value for contact width is of practical
interest. As the load, P = 4080lbs generates a contact width ‘2a’ of 0.02in, this value of
load is selected.
Material Properties and Model Characteristics
The finite element model requires modulus of elasticity ‘E’, modulus of rigidity
’G’ and Poisson’s ratio ‘ν’ as the only input material property parameters in ANSYS. By
defining these parameters we can model a single crystal anisotropic substrate. The length,
width and height of the model are taken as 0.25in. The dimensions of the model have
59
been selected accordingly due to the limitation to the maximum number of nodes in
ANSYS.
Anisotropic (FCC single crystal) substrate cannot be modeled in 2-D as it can
have out of plane stresses induced because of shear coupling. Thus, the plane strain
conditions are not valid. Hence it is modeled in 3-D that increases the number of nodes in
the model. The coordinate transformations are possible only in the 3-D model.
Figure 5-4 Dimensions of the anisotropic contact model
The element chosen in this study is SOLID 45. ANSYS reference manual [43]
gives the description for the elements. SOLID 45 (Fig 5.5) is used for the threedimensional modeling of solid structures. The element is defined by eight nodes having
three degrees of freedom at each node: translations in the nodal x, y and z directions.
60
Figure 5-5 ANSYS SOLID 45 element Source: ANSYS 6.1 Elements Reference, 2002
The area near the contact is meshed separately for accurate results (Fig.5.6) as
compared to the whole body in order to restrict the maximum number of nodes in the
body. Fig. 5-6 shows the front view of the highly meshed contact region and Fig. 5-7
shows the fully meshed anisotropic single crystal substrate in 3-D.
Figure 5-6 Refined meshing in the contact zone
61
Figure 5-7 Meshed anisotropic FCC single crystal substrate
CHAPTER 6
RESULTS AND DISCUSSION
The main purpose of this study is to analyze subsurface stresses in an anisotropic
(FCC single crystal) substrate subjected to contact loading under various crystallographic
orientations. The stresses are further used to evaluate the RSS values to determine the
dominant and activated slip planes. The RSS values are evaluated on the primary
octahedral slip systems in the subsurface region as a function of radial and angular
position as shown in Fig 6.1. Each of the five crystallographic orientations examined
show different slip activations at different locations, emphasizing the effect of the
material’s anisotropy. Each orientation is discussed individually. Further the finite
element solution is compared for orientations 1 and 4 with the analytical results.
2a
Figure 6-1 Schematic of the polar coordinate system used in the subsurface contact
region
62
63
Crystallographic Orientation 1
In this crystallographic orientation the specimen coordinate axes and the material
coordinate axes are the same given by x [100], y [010], and z [001]. Figures 6.2-6.4 show
that the maximum RSS at any location is τ2 = τ10 = 7.496*104 psi at r = 0.8*a and 400
(leading edge of the contact subsurface). Results are presented for the twelve primary
RSS values for r = 0.3*a, r = 0.8*a and r = 3*a and from 00 to 1800. The dominant slip
system with the maximum RSS varies with radial and angular position. Notice that for a
given angle the dominant slip system (and often the other activated systems) is not
constant over the range of the radii. As the state of stress changes away from the contact
region, the RSS also changes.
In general the slip systems are highly variable throughout the RSS field though
some angles do maintain a single dominant slip system for all radii; τ2 and τ10 at 400-700
and τ3, τ5, τ7 and τ12 at 1000 -1100. Overall the RSS field is dominated by τ2, τ10, τ3, τ5, τ7
and τ12 as well as τ6 and τ8 in some areas.
Table 6.1 gives a detail explanation of the activated slip systems for the
orientation 1. The activated slip systems are known from the Fig 6.2-6.4. The resolved
shear stress values, which are higher than the material’s critical resolved shear stress
value of 47ksi, gets activated. τ2 and τ10 are activated for all the angles at r =0.3*a and
from the angles 00-900 for r = 0.8*a. At r = 3*a no slip systems are activated.
RSS Vs Theta - Specimen 1 (r = 0.3*a)
4
4
6.273 × 10 7 . 10
τ(θ)0,0
τ(θ)1,0
4
6 . 10
τ(θ)2,0
τ(θ)3,0
CRSS value = 47ksi
4
5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
4 . 10
τ(θ)6,0
4
3 . 10
64
τ(θ)7,0
τ(θ)8,0
τ(θ)9,0
4
2 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0 1 . 10 4
213.523
0
0.5
1
0.175
Figure 6-2 Orientation 1 primary resolved shear stresses; r = 0.3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 1 (r = 0.8*a)
4 . 4
7.496 × 10 8 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
7 . 10
4
6 . 10
4
5 . 10
4
4 . 10
4
3 . 10
4
2 . 10
4
1 . 10
4
τ(θ)3,0
RSS (psi)
τ(θ)4,0
CRSS value = 47ksi
τ(θ)5,0
τ(θ)6,0
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
65
τ(θ)8,0
109.702
0
0.5
1
0.175
Figure 6-3 Orientation 1 primary resolved shear stresses; r = 0.8*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS vs Theta - Specimen 1 (r = 3*a)
4
. 4
3.321 × 10 3.5 10
τ(θ)0,0
τ(θ)1,0
4
3 . 10
τ(θ)2,0
τ(θ)3,0
4
2.5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
2 . 10
τ(θ)6,0
τ(θ)7,0
4
1.5 . 10
τ(θ)9,0
66
τ(θ)8,0
4
1 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
5000
48.116
0
0.5
1
0.175
Figure 6-4 Orientation 1 primary resolved shear stresses; r = 3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
Table 6-1 Orientation 1 activated slip system sectors
Activated Slip System Sectors
Orientation 1 - x [100] y [010] z [001]r = 0.3*a
r = 0.8*a
Sector
θ
τ
Activated Slip Systems
θ
τ
1
0-55
τ2 and τ10
(111)[0-11], (-1-11)[011]
0-40
τ2 and τ10
τ2, τ10,
(111)[0-11], (-1-11)[011],
(111)[1-10], (-11-1)[110],
(1-1-1)[110], (-1-11)[110]
40-46
2
55-145
τ3, τ5,
τ7 τ12
τ2, τ10,
τ4, τ9
τ2, τ10,
3
145-180
τ2 and τ10
(111)[0-11], (-1-11)[011]
46-55
τ4, τ9,
τ3, τ5,
τ2, τ10,
4
55-90
τ3, τ5,
τ7 τ12
5
90-95
τ3 , τ5 ,
τ7 τ12
τ3, τ5,
6
7
95-142
142-180
Activated Slip Systems
(111)[0-11], (-1-11)[011]
(111)[0-11], (-1-11)[011],
(-11-1)[10-1], (1-1-1)[101]
(111)[0-11], (-1-11)[011],
(-11-1)[10-1], (1-1-1)[101],
(111)[1-10], (-11-1)[110],
(1-1-1)[110], (-1-11)[110]
(111)[0-11], (-1-11)[011],
(111)[1-10], (-11-1)[110],
(1-1-1)[110], (-1-11)[110]
(111)[1-10], (-11-1)[110],
(1-1-1)[110], (-1-11)[110]
τ6 , τ8
(111)[1-10], (-11-1)[110],
(1-1-1)[110], (-1-11)[110],
(-11-1)[011], (1-1-1)[0-11]
τ6 and τ8
(-11-1)[011], (1-1-1)[0-11]
τ7, τ12,
No slip
systems are
activated
67
τ7 τ12
r = 3*a
68
Crystallographic Orientation 2
Orientation 2 is the off-axes case of anisotropy where the specimens coordinate
axes and the material coordinate axes are not the same and where δ = 150, γ = θ = 0 (Fig
5.1) (Case 17). Figures 6.5-6.7 shows that the maximum RSS at any location is τ3 = τ5 =
τ7 = τ12 = 7.304*104 psi at r = 0.8*a and 1000. Notice that this value is slightly lower than
the first case value but occurs at a different location. Results are again presented for the
twelve primary RSS values for r = 0.3*a to r = 0.8*a and r = 3*a and from 00 to 1800. The
dominant slip system with the maximum RSS varies with radial and angular position.
Notice that again for a given angle the dominant system (and often the other activated
systems) is not constant over the range of the radii.
In general the slip systems are highly variable throughout the RSS field though
some angles do maintain a single dominant slip system for all radii; τ2 and τ10 at 450-700.
Overall the RSS field is dominated by τ2 and τ10 as well as τ6 and τ8 in some areas.
Table 6.2 gives a detail explanation of the activated slip systems for the
orientation 2. The activated slip systems are known from the Fig 6.5-6.7. The resolved
shear stress values, which are higher than the material’s critical resolved shear stress
value of 47ksi, gets activated. τ2 and τ10 are activated for all the angles at r =0.3*a and
from the angles 00-950 for r = 0.8*a. At r = 3*a no slip systems are activated.
RSS vs Theta - Specimen 2 (r = 0.3*a)
4
6.374 × 10 7 . 10
4
τ(θ)0,0
τ(θ)1,0
4
6 . 10
τ(θ)2,0
τ(θ)3,0
CRSS value = 47ksi
4
5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
4 . 10
τ(θ)6,0
τ(θ)7,0
4
3 . 10
τ(θ)9,0
69
τ(θ)8,0
4
2 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
1 . 10
556.771
0
0.5
1
0.175
Figure 6-5 Orientation 2 primary resolved shear stresses; r = 0.3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS vs Theta - Specimen 2 (r = 0.8*a)
4
4 .
7.304 × 10 8 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
4
7 . 10
4
6 . 10
τ(θ)3,0
RSS (psi)
τ(θ)4,0
4
5 . 10
CRSS value = 47ksi
τ(θ)5,0
τ(θ)6,0
4
4 . 10
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
3 . 10
70
τ(θ)8,0
4
2 . 10
4
1 . 10
174.588
0
0.5
1
0.175
Figure 6-6 Orientation 2 primary resolved shear stresses; r = 0.8*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
3.454 × 10
4
RSS vs Theta - Specimen 2 (r = 3*a)
4
3.5 . 10
τ(θ)0,0
τ(θ)1,0
4
3 . 10
τ(θ)2,0
τ(θ)3,0
4
2.5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
2 . 10
τ(θ)6,0
τ(θ)7,0
4
1.5 . 10
τ(θ)9,0
71
τ(θ)8,0
4
1 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
5000
407.417
0
0.5
1
0.175
Figure 6-7 Orientation 2 primary resolved shear stresses; r = 3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
Table 6-2 Orientation 2 activated slip system sectors
Activated Slip System Sectors
Orientation 2 - δ = 150 γ = θ = 0
r = 0.3*a
r = 0.8*a
Sector
θ
1
0-18
2
18-70
τ
τ2, τ10,
τ1, τ11
τ2 and
τ10
τ2, τ10,
70-150
τ3, τ5,
τ7, τ12
(111)[0-11], (-1-11)[011]
(111)[0-11], (-111)[011], (111)[1-10], (11-1)[110]
(1-1-1)[110], (-111)[110]
θ
0-35
35-55
τ
Activated Slip Systems
τ2, τ10,
τ1, τ11
(111)[0-11], (-1-11)[011],
(111)[10-1], (-1-11)[101]
τ2 and τ10
(111)[0-11], (-1-11)[011]
τ2, τ10,
55-90
τ3 τ5 ,
τ7, τ12
τ2, τ10,
4
150-180
τ2 and
τ10
(111)[0-11], (-1-11)[011]
90-95
τ3, τ5,
τ7, τ12,
τ6 , τ8
τ3, τ5,
5
95-180
τ7, τ12,
τ6 , τ8
(111)[0-11], (-111)[011], (111)[1-10],
(-11-1)[110]
(1-1-1)[110], (-111)[110]
(111)[0-11], (-111)[011], (111)[1-10],
(-11-1)[110]
(1-1-1)[110], (-111)[110], (-11-1)[011], (11-1)[0-11]
(111)[1-10], (-111)[110]
(1-1-1)[110], (-111)[110], (-11-1)[011], (11-1)[0-11]
72
3
Activated Slip Systems
(111)[0-11], (-111)[011], (111)[10-1], (1-11)[101]
r = 3*a
No slip
systems are
activated
73
Crystallographic Orientation 3
Orientation 3 is the case of anisotropy where δ = γ = 10.610, θ = 00 (Fig 5.1) (Case
19). Figures 6.8-6.10 shows that the maximum RSS at any location is τ5 = 7.635*104 psi
at r = 0.8*a and 900. Notice that this value is slightly higher than the first case value but
occurs at a different location. Results are again presented for the twelve primary RSS
values for r = 0.3*a, r = 0.8*a and r = 3*a and from 00 to 1800. The dominant slip system
with the maximum RSS varies with radial and angular position. Notice that again for a
given angle the dominant system (and often the other activated systems) is not constant
over the range of the radii. The dominant slip system changes one or more times for a
particular value of theta.
In this case a single dominant slip system for all radii is only maintained at
approximately 700 by τ2. In general the slip systems are highly variable throughout the
RSS field. Overall the RSS field is dominated by τ2, τ5, τ6 and τ10.
Table 6.3 gives a detail explanation of the activated slip systems for the
orientation 3. The activated slip systems are known from the Fig 6.8-6.10. The resolved
shear stress values, which are higher than the material’s critical resolved shear stress
value of 47ksi, gets activated. τ2 and τ10 are activated for all the angles at r =0.3*a. τ2 is
activated from 100 to 1000, τ10 from 00 to 960, τ5 from 520-1800 and τ6 from 850-1800 for r
= 0.8*a. At r = 3*a no slip systems are activated.
RSS Vs Theta - Specimen 3 (r = 0.3*a)
4
6.829 × 10 7 . 10
4
τ(θ)0,0
τ(θ)1,0
4
6 . 10
τ(θ)2,0
τ(θ)3,0
CRSS value = 47ksi
4
5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
4 . 10
τ(θ)6,0
τ(θ)7,0
4
3 . 10
τ(θ)9,0
74
τ(θ)8,0
4
2 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
1 . 10
513.17
0
0.5
1
0.175
Figure 6-8 Orientation 3 primary resolved shear stresses; r = 0.3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 3 (r = 0.8*a)
4 . 4
7.635 × 10 8 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
7 . 10
4
6 . 10
4
5 . 10
4
4 . 10
4
3 . 10
4
2 . 10
4
1 . 10
4
τ(θ)3,0
RSS (psi)
τ(θ)4,0
CRSS value = 47ksi
τ(θ)5,0
τ(θ)6,0
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
1.431 × 10
75
τ(θ)8,0
3
0
0.5
1
0.175
Figure 6-9 Orientation 3 primary resolved shear stresses; r = 0.8*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 3 (r = 3*a)
4
. 4
3.444 ×10 3.5 10
τ(θ)0, 0
τ(θ)1, 0
3 . 10
4
τ(θ)2, 0
τ(θ)3, 0
4
2.5 . 10
RSS (psi)
τ(θ)4, 0
τ(θ)5, 0
2 . 10
4
τ(θ)6, 0
τ(θ)7, 0
4
1.5 . 10
76
τ(θ)8, 0
τ(θ)9, 0
1 . 10
4
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
5000
8.03
0
0.5
1
0.175
Figure 6-10 Orientation 3 primary resolved shear stresses; r = 3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
Table 6-3 Orientation 3 activation slip system sectors
Activated Slip System Sectors
Orientation 3 - δ = γ = 10.610 θ = 0
r = 0.3*a
r = 0.8*a
θ
τ
Activated Slip Systems
θ
τ
Activated Slip Systems
1
0-18
τ2, τ10,
τ1, τ11
(111)[0-11], (-111)[011], (111)[10-1], (1-11)[101]
0-10
τ10, τ1,
τ11
(-1-11)[011], (111)[10-1],
(-1-11)[101]
2
18-25
τ2, τ10,
τ1
(111)[0-11], (-111)[011], (111)[10-1]
10-35
τ2, τ10,
τ1, τ11
(111)[0-11], (-1-11)[011],
(111)[10-1], (-1-11)[101]
3
25-70
τ2, τ10
(111)[0-11], (-1-11)[011
35-42
τ2, τ10,
τ1
(111)[0-11], (-1-11)[011],
(111)[10-1]
4
70-75
τ2, τ10,
τ12
(111)[0-11], (-111)[011], (-1-11)[1-10]
42-50
τ2, τ10
(111)[0-11], (-1-11)[011]
75-85
τ2, τ10,
τ12, τ5
(111)[0-11], (-111)[011], (-1-11)[1-10],
(-11-1)[110]
50-52
τ2, τ10,
τ12
(111)[0-11], (-1-11)[011],
(-1-11)[1-10]
85-90
τ2, τ10,
τ12, τ5,
τ3
(111)[0-11], (-111)[011], (-1-11)[1-10],
(-11-1)[110], (111)[1-10]
52-60
τ2, τ10,
τ12, τ5
(111)[0-11], (-1-11)[011],
(-1-11)[110], (-11-1)[110],
90-125
τ2, τ10,
τ12, τ5,
τ3 , τ7
(111)[0-11], (-111)[011], (-1-11)[1-10],
(-11-1)[110], (111)[110], (1-1-1)[110]
60-62
τ2, τ10,
τ12, τ5,
τ3
(111)[0-11], (-1-11)[011],
(-1-11)[110], (-11-1)[110],
(111)[1-10]
125-145
τ2, τ10,
τ12, τ5,
τ3
(111)[0-11], (-111)[011], (-1-11)[1-10],
(-11-1)[110], (111)[1-10]
62-85
τ2, τ10,
τ12, τ5,
τ3 , τ7
(111)[0-11], (-1-11)[011],
(-1-11)[110], (-11-1)[110],
(111)[1-10], (1-1-1)[110]
5
6
7
8
No slip
systems are
activated
77
Sector
r = 3*a
Table 6-3. Continued
9
145-155
τ2, τ10,
τ5
(111)[0-11], (-111)[011], (-11-1)[110
85-88
τ2, τ10,
τ12, τ5,
τ3 , τ7 ,
τ6
10
155-180
τ2, τ10
(111)[0-11], (-1-11)[011]
88-96
τ2, τ10,
τ12, τ5,
τ3 , τ7 ,
τ6 , τ8
11
96-100
τ2, τ12,
τ5 , τ3 ,
τ7 , τ6 ,
τ8
100-148
τ6 , τ8
13
148-155
τ12, τ5,
τ3 , τ6 ,
τ8
14
155-170
15
170-180
τ5 , τ3 ,
τ6 , τ8
τ5 , τ3 ,
τ6
(111)[0-11], (-1-11)[011],
(-1-11)[110], (-11-1)[110],
(111)[1-10], (1-1-1)[110],
(-11-1)[011], (1-1-1)[0-11]
(111)[0-11], (-1-11)[110],
(-11-1)[110], (111)[1-10],
(1-1-1)[110], (-11-1)[011],
(1-1-1)[0-11]
(-1-11)[110], (-11-1)[110],
(111)[1-10], (1-1-1)[110],
(-11-1)[011], (1-1-1)[0-11]
(-1-11)[110], (-11-1)[110],
(111)[1-10], (-11-1)[011],
(1-1-1)[0-11]
(-11-1)[110], (111)[1-10],
(-11-1)[011], (1-1-1)[0-11]
(-11-1)[110], (111)[1-10],
(-11-1)[011]
78
12
τ12, τ5,
τ3 , τ7 ,
(111)[0-11], (-1-11)[011],
(-1-11)[110], (-11-1)[110],
(111)[1-10], (1-1-1)[110],
(-11-1)[011]
79
Crystallographic Orientation 4
Orientation 4 is again the off-axes case of anisotropy where δ = γ = 10.610, θ =
400 (Fig 5.1). Figures 6.11-6.13 shows that the maximum RSS at any location is τ1 =
1.189*105 psi at r = 0.3*a and 00 which is at the surface. Notice that this value is
comparatively higher than the previous case maximum values and occurs totally at a
different radius and angle. Results are again presented for the twelve primary RSS values
for r = 0.3*a, r = 0.8*a and r = 3*a and from 00 to 1800. The dominant slip system with
the maximum RSS varies with radial and angular position. Notice that again for a given
angle the dominant system (and often the other activated systems) is not constant over the
range of the radii.
In this case a single dominant slip system for all radii is maintained from 750 to
1100 only by τ5. In general the slip systems are highly variable throughout the RSS field.
Overall the RSS field is dominated by τ1 and τ5 and somewhat by τ2 and τ12.
Table 6.4 gives a detail explanation of the activated slip systems for the
orientation 3. The activated slip systems are known from the Fig 6.11-6.13. The resolved
shear stress values, which are higher than the material’s critical resolved shear stress
value of 47ksi, gets activated. τ2 is activated for all the angles at r = 0.3*a and τ5 is
activated from 600-1580 for the same radius. τ2 is activated from 00 to 640, τ5 from 460 to
1800 and τ12 from 600-1800 for r = 0.8*a. At r = 3*a no slip systems are activated.
1.028 × 10
RSS Vs Theta - Specimen 4 (r = 0.3*a)
5 1.2 . 10 5
τ(θ)0,0
τ(θ)1,0
5
1 . 10
τ(θ)2,0
τ(θ)3,0
RSS (psi)
τ(θ)4,0
4
8 . 10
τ(θ)5,0
τ(θ)6,0
4
6 . 10
CRSS value = 47ksi
τ(θ)7,0
4
4 . 10
80
τ(θ)8,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
2 . 10
168.964
0
0.5
1
0.175
Figure 6-11 Orientation 4 primary resolved shear stresses; r = 0.3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 4 (r = 0.8*a)
4 . 5
8.339 × 10 1 10
τ(θ)0, 0
τ(θ)1, 0
τ(θ)2, 0
8 . 10
4
6 . 10
4
τ(θ)3, 0
RSS (psi)
τ(θ)4, 0
τ(θ)5, 0
CRSS value = 47ksi
τ(θ)6, 0
τ(θ)7, 0
4
2 . 10
4
81
τ(θ)8, 0
4 . 10
τ(θ)9, 0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
1.413 × 10
3
0
0.5
1
0.175
Figure 6-12 Orientation 4 primary resolved shear stresses; r = 0.8*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
3.518 × 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
4
RSS Vs Theta - Specimen 4 (r = 3*a)
4
4 . 10
4
3.5 . 10
4
3 . 10
τ(θ)3,0
RSS (psi)
τ(θ)4,0
4
2.5 . 10
τ(θ)5,0
τ(θ)6,0
4
2 . 10
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
1.5 . 10
82
τ(θ)8,0
4
1 . 10
5000
27.379
0
0.5
1
0.175
Figure 6-13 Orientation 4 primary resolved shear stresses; r = 3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
Table 6-4 Orientation 4 activated slip system sectors
Activated Slip System Sectors
Orientation 4- δ = γ = 10.610 θ = 400
r = 0.3*a
r = 0.8*a
θ
τ
Activated Slip Systems
θ
1
0-15
τ 1, τ 2,
τ10, τ11,
τ7
(111)[10-1], (111)[0-11],
(-1-11)[011], (-1-11) [101],
(1-1-1)[110]
0-10
2
15-23
τ 1, τ 2,
τ10, τ11
(111)[10-1], (111)[0-11],
(-1-11)[011], (-1-11) [101]
10-22
3
23-35
τ 1, τ 2,
τ10
(111)[10-1], (111)[0-11],
(-1-11)[011]
22-35
Sector
r = 3*a
τ
Activated Slip Systems
τ 1, τ 2,
(111)[10-1], (111)[0-11],
(-1-11)[011], (1-1-1)[110]
τ10, τ7
τ 1, τ 2,
τ10
(111)[10-1], (111)[0-11],
(-1-11)[011]
τ 1, τ 2
(111)[10-1], (111)[0-11]
35-60
τ 1, τ 2
(111)[10-1], (111)[0-11]
35-40
τ2
(111)[0-11]
5
60-90
τ 2, τ 5
(111)[0-11], (-11-1)[110]
40-46
τ 2, τ 4
(111)[0-11], (-11-1)[10-1]
6
90-140
τ 2, τ 5,
τ12
(111)[0-11], (-11-1)[110],
(-1-11)[1-10]
46-60
τ 2, τ 4,
(111)[0-11], (-11-1)[10-1],
(-11-1)[110]
7
140-155
τ 2, τ 5
(111)[0-11], (-11-1)[110]
60-64
8
155-158
τ 1, τ 2,
τ5
(111)[10-1], (111)[0-11],
(-11-1)[110]
64-72
τ5
τ2, τ4,
τ5, τ12
τ4,
τ5, τ12
(111)[0-11], (-11-1)[10-1],
(-11-1)[110], (-1-11)[1-10]
(-11-1)[10-1],
(-11-1)[110], (-1-11)[1-10]
83
4
No slip
systems are
activated
Table 6-4. Continued
9
158-180
τ 1, τ 2,
τ10
(111)[10-1], (111)[0-11],
(-1-11)[011]
72-85
85-100
11
100-115
12
115-120
13
120-150
14
150-160
15
160-180
τ5, τ12
τ 3, τ 5,
τ12
τ5, τ12,
τ11
τ5, τ12,
τ11, τ6
τ5, τ12,
τ 6, τ 8
τ5, τ12,
(111) [1-10], (-11-1)[10-1],
(-11-1)[110], (-1-11)[1-10]
(111) [1-10], (-11-1)[110],
(-1-11)[1-10]
(-11-1)[110], (-1-11)[1-10],
(-1-11)[101]
(-11-1)[110], (-1-11)[1-10],
(-1-11)[101], (-11-1)[011]
(-11-1)[110], (-1-11)[1-10],
(-11-1)[011], (1-1-1)[0-11]
τ8
(-11-1)[110], (-1-11)[1-10],
(1-1-1)[0-11]
τ5, τ12
(-11-1)[110], (-1-11)[1-10]
84
10
τ3, τ4,
85
Crystallographic Orientation 5
Orientation 5 is the last case considered and is again the off-axes case of
anisotropy where δ = γ = 00, θ = 400 (Fig 5.1). Figures 6.14-6.16 shows that the
maximum RSS at any location is τ10 = 7.7*104 psi at r = 0.8*a and 170. Notice that this
value is higher than all the previous case maximum values except the fourth case. Results
are again presented for the twelve primary RSS values from 0.3*a to 3*a and from 00 to
1800 for each radius as was presented till now. The dominant slip system with the
maximum RSS varies with radial and angular position. Notice that again for a given
angle the dominant system (and often the other activated systems) is not constant over the
range of the radii.
In this case there is no single dominant slip system for all radii and some
particular angles. So the slip systems are highly variable throughout the RSS field.
Overall the RSS field is dominated by τ10 for smaller radius of 0.3*a and for higher radii
the domination is by multiple τ.
Table 6.5 gives a detail explanation of the activated slip systems for the
orientation 3. The activated slip systems are known from the Fig 6.14-6.16. The resolved
shear stress values, which are higher than the material’s critical resolved shear stress
value of 47ksi, gets activated. τ10 is activated for the angles 00-1200 and then 1420-1800.at
r = 0.3*a and τ2 is activated from 00-1200 and then 1470-1800for the same radius. τ3 is
activated from 400 to 1490, τ7 from 400 to 1480, τ5 from 420-1530 and τ12 from 420-1480
for r = 0.8*a. At r = 3*a no slip systems are activated.
RSS Vs Theta - Specimen 5 (r = 0.3*a)
4
4 .
7.411 × 10 8 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
4
7 . 10
4
6 . 10
τ(θ)3,0
RSS (psi)
τ(θ)4,0
CRSS value = 47ksi
4
5 . 10
τ(θ)5,0
τ(θ)6,0
4
4 . 10
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
4
3 . 10
86
τ(θ)8,0
4
2 . 10
4
1 . 10
694.093
0
0.5
1
0.175
Figure 6-14 Orientation 5 primary resolved shear stresses; r = 0.3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 5 (r = 0.8*a)
4 . 4
7.7 × 10 8 10
τ(θ)0,0
τ(θ)1,0
τ(θ)2,0
7 . 10
4
6 . 10
4
5 . 10
4
4 . 10
4
3 . 10
4
2 . 10
4
1 . 10
4
τ(θ)3,0
RSS (psi)
τ(θ)4,0
τ(θ)5,0
τ(θ)6,0
CRSS value = 47ksi
τ(θ)7,0
τ(θ)9,0
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
87
τ(θ)8,0
252.609
0
0.5
1
0.175
Figure 6-15 Orientation 5 primary resolved shear stresses; r = 0.8*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
RSS Vs Theta - Specimen 5 (r = 3*a)
4
. 4
3.418 × 10 3.5 10
τ(θ)0,0
τ(θ)1,0
4
3 . 10
τ(θ)2,0
τ(θ)3,0
4
2.5 . 10
RSS (psi)
τ(θ)4,0
τ(θ)5,0
4
2 . 10
τ(θ)6,0
τ(θ)7,0
4
1.5 . 10
τ(θ)9,0
88
τ(θ)8,0
4
1 . 10
τ ( θ ) 10 , 0
τ ( θ ) 11 , 0
5000
11.974
0
0.5
1
0.175
Figure 6-16 Orientation 5 primary resolved shear stresses; r = 3*a
1.5
θ
Theta (rad)
2
2.5
3
2.967
Table 6-5 Orientation 5 activated slip system sectors
Activated Slip System Sectors
Orientation 5 - δ = γ = 0, θ = 400
r = 0.3*a
Sector
θ
1
0-18
2
20-45
Activated Slip Systems
θ
τ 1, τ 2,
(111)[10-1], (111)[0-11],
(-1-11)[011], (-1-11)[101]
0-27
τ10, τ11
τ 2,
τ10, τ11
τ2, τ10
(111)[0-11],
(-1-11)[011], (-1-11)[101]
(111)[0-11], (-1-11)[011]
τ
Activated Slip Systems
τ 1, τ 2,
(111)[10-1], (111)[0-11],
(-1-11)[011], (-1-11)[101]
τ10, τ11
τ 1, τ 2,
27-30
30-33
τ10, τ11,
τ9
τ2, τ10,
τ11, τ9,
τ4
4
45-48
τ2, τ10,
τ7
τ2, τ10,
5
48-52
τ 7, τ 3,
τ12
τ2, τ10,
6
52-120
τ 7, τ 3,
τ12, τ5
(111)[0-11], (-1-11)[011],
(1-1-1)[110]
(111)[0-11], (-1-11)[011],
(1-1-1)[110], (111) [1-10],
(-1-11)[1-10]
(111)[0-11], (-1-11)[011],
(1-1-1)[110], (111) [1-10],
(-1-11)[1-10], (-11-1)[110]
33-40
τ2, τ10,
τ 9, τ 4
τ2, τ10,
40-42
τ 9, τ 4,
τ 3, τ 7,
τ2, τ10,
42-75
τ 9, τ 4,
τ 3, τ 7,
τ12, τ5
r = 3*a
(111)[10-1], (111)[0-11],
(-1-11)[011], (-1-11)[101],
(1-1-1)[101]
(111)[0-11], (-1-11)[011],
(-1-11)[101], (1-1-1)[101]
(-11-1)[10-1]
(111)[0-11], (-1-11)[011],
(1-1-1)[101](-11-1)[10-1]
(111)[0-11], (-1-11)[011],
(1-1-1)[101](-11-1)[10-1],
(111) [1-10], (1-1-1)[110]
(111)[0-11], (-1-11)[011],
(1-1-1)[101](-11-1)[10-1],
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110]
No slip
systems
are
activated
89
3
18-20
τ
r = 0.8*a
Table 6-5. Continued
τ 2, τ 9,
7
120-142
τ 7, τ 3,
τ12, τ5
(1-1-1)[110], (111) [1-10],
(-1-11)[1-10], (-11-1)[110]
75-78
τ 4, τ 3,
τ7, τ12,
τ5
τ10,
8
142-147
τ 7, τ 3,
τ12, τ5
τ2, τ10,
9
147-150
τ 7, τ 3,
τ12, τ5
10
150-152
τ7, τ12,
τ5
(111)[0-11], (-1-11)[011],
(1-1-1)[110], (111) [1-10],
(-1-11)[1-10], (-11-1)[110]
(111)[0-11], (-1-11)[011],
(1-1-1)[110], (-1-11)[110], (-11-1)[110]
τ 9, τ 4,
78-90
τ 3, τ 7,
τ12, τ5
τ 9, τ 4,
90-93
τ 3, τ 7,
τ12, τ5,
τ1,τ11
τ 3, τ 7,
93-120
τ12, τ5,
τ1, τ11
τ 3, τ 7,
11
152-155
τ2, τ10,
τ7, τ12
(111)[0-11], (-1-11)[011],
(1-1-1)[110], (-1-11)[1-10]
120-124
τ12, τ5,
τ1,τ11,
τ6
τ 3, τ 7,
12
155-180
τ2, τ10
(111)[0-11], (-1-11)[011
124-148
τ12, τ5,
τ1, τ11,
τ 6, τ 8
(1-1-1)[101], (-11-1)[10-1],
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110]
(1-1-1)[101], (-11-1)[10-1],
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110],
(111)[10-1], (-1-11)[101]
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110],
(111)[10-1], (-1-11)[101]
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110],
(111)[10-1], (-1-11)[101],
(-11-1)[011]
(111) [1-10], (1-1-1)[110],
(-1-11)[1-10], (-11-1)[110],
(111)[10-1], (-1-11)[101],
(-11-1)[011], (1-1-1)[0-11]
90
τ2, τ10,
(-1-11)[011],
(1-1-1)[110], (111) [1-10],
(-1-11)[1-10], (-11-1)[110]
(111)[0-11], (1-1-1)[101],
(-11-1)[10-1], (111) [1-10],
(1-1-1)[110], (-1-11)[1-10],
(-11-1)[110]
Table 6-5. Continued
τ3, τ12,
13
148-149
τ 5, τ 1,
τ11, τ6,
τ8
τ12, τ5,
14
149-153
τ1, τ11,
τ 6, τ 8
153-163
16
163-168
17
168-180
τ1, τ11,
τ 6, τ 8
τ11, τ6,
(-1-11)[1-10], (-111)[110], (111)[10-1], (-111)[101],
(-11-1)[011], (1-1-1)[0-11]
(111)[10-1], (-1-11)[101],
(-11-1)[011], (1-1-1)[0-11]
τ8
(-1-11)[101], (-11-1)[011],
(1-1-1)[0-11]
τ 6, τ 8
(-11-1)[011], (1-1-1)[0-11]
91
15
(111) [1-10], (-1-11)[1-10],
(-11-1)[110], (111)[10-1],
(-1-11)[101], (-11-1)[011],
(1-1-1)[0-11]
92
Comparison of FEM Solution with the Analytical Results for Case 1 Neglecting
Tangential Traction
Figure 6-17 Contour plot for sigma (x) -Analytical solution
Figure 6-18 Contour plot for sigma (x) -FEM solution
93
Figure 6-19 Contour plot for sigma (y) -Analytical solution
Figure 6-20 Contour plot for sigma (y)-FEM solution
94
Figure 6-21 Contour plot for tau (xy)-Analytical solution
Figure 6-22 Contour plot for tau (xy)-FEM solution
95
Figure 6-23 Contour plot for sigma (z)-Analytical solution
Figure 6-24 Contour plot for sigma (x)-FEM solution
Comparison of FEM Solution with the Analytical Results
-2.50E+05
-2.00E+05
-1.50E+05
-1.00E+05
-5.00E+04
0
0.00E+00
5.00E+04
-0.005
-0.01
-0.015
-0.02
-0.025
σx psi
Sx-analytical
Sx-numerical coarse mesh
Figure 6-25 Comparison of stresses in the X-direction for orientation 1
Sx-numerical fine mesh
96
Distance along the subsurface depth (in)
Case1
-3.50E+05
-3.00E+05
-2.50E+05
-2.00E+05
-1.50E+05
-1.00E+05
-5.00E+04
0.00E+00
-0.005
-0.01
-0.015
-0.02
-0.025
σy psi
Sy-analytical
Sy-numerical-coarse mesh
Figure 6-26 Comparison of stresses in the Y-direction for orientation 1
Sy numerical fine mesh
97
Distance along the subsurface depth
(in)
0
-2.00E+05
-1.50E+05
-1.00E+05
-5.00E+04
-0.005
-0.01
-0.015
-0.02
-0.025
σz
Sz-analytical
Sz-numerical-coarse mesh
Figure 6-27 Comparison of stresses in the Z-direction for orientation 1
Sz numerical fine mesh
98
Distance along the subsurface depth (in)
-2.50E+05
0
0.00E+00
-6.00E+04
-5.00E+04
-4.00E+04
-3.00E+04
Distance along the subsurface depth (in)
-7.00E+04
-2.00E+04
-1.00E+04
0
0.00E+00
-0.005
-0.01
-0.015
99
-0.02
-0.025
τxy psi
Txy-analytical
Txy-numerical-coarse mesh
Figure 6-28 Comparison of shear stresses in the XY-direction for orientation 1
Txy-numerical fine mesh
-2.00E+03
-1.50E+03
-1.00E+03
5.00E+02
1.00E+03
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.016
-0.018
-0.02
τ yz psi
Tyz-analytical
Syz-numerical-coarse mesh
Figure 6-29 Comparison of shear stresses in the YZ-direction for orientation 1
Syz-numerical fine mesh
100
Distance along the subsurface depth
(in)
-2.50E+03
0
-5.00E+02 0.00E+00
-0.002
-0.005
-0.01
-0.015
101
Distance along the subsurface
depth (in)
0
-1.20E+ -1.00E+ -8.00E+ -6.00E+ -4.00E+ -2.00E+ 0.00E+ 2.00E+ 4.00E+ 6.00E+ 8.00E+
03
03
02
02
02
02
00
02
02
02
02
-0.02
-0.025
τ xz psi
Sxz-analytical
Sxz-numerical-coarse mesh
Figure 6-30 Comparison of shear stresses in the XZ-direction for orientation 1
Sxz-numerical fine mesh
-3.00E+05
-2.50E+05
-2.00E+05
-1.50E+05
-1.00E+05
0
-5.00E+04
0.00E+00
-0.002
5.00E+04
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.016
-0.018
-0.02
σx psi
Sx analytical
Sx numerical coarse mesh
Figure 6-31 Comparison of stresses in the X-direction for orientation 4
Sx numerical fine mesh
102
distance along the subsurface depth (in)
Case 4
-2.50E+05
-2.00E+05
-1.50E+05
-1.00E+05
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.016
-0.018
-0.02
σy psi
Sy analytical
Sy numerical coarse mesh
Figure 6-32 Comparison of stresses in the Y-direction for orientation 4
Sy numerical fine mesh
103
distance along the subsurface depth
(in)
-3.00E+05
0
-5.00E+04
0.00E+00
-0.002
-2.00E+05
-1.50E+05
-1.00E+05
-5.00E+04
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.016
-0.018
-0.02
σz psi
Sz analytical
Sz numerical coarse mesh
Figure 6-33 Comparison of stresses in the Z-direction for orientation 4
Sz numerical fine mesh
104
distance along the subsurface depth
(in)
-2.50E+05
0
0.00E+00
-0.002
-6.00E+04
-4.50E+04
-3.00E+04
distance along the subsurface
depth (in)
-7.50E+04
0
-1.50E+04
0.00E+00
-0.002
1.50E+04
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.018
-0.02
τxy psi
Sxy analytical
Sxy numerical coarse mesh
Figure 6-34 Comparison of shear stresses in the XY-direction for orientation 4
Sxy numerical fine mesh
105
-0.016
1.50E+04
3.00E+04
4.50E+04
6.00E+04
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-0.016
106
distance along the subsurface depth (in)
-1.50E+04
0
0.00E+00
-0.002
-0.018
-0.02
τ yz psi
Syz analytical
Syz numerical coarse mesh
Figure 6-35 Comparison of shear stresses in the YZ-direction for orientation 4
Syz numerical fine mesh
5.00E+03
1.00E+04
1.50E+04
2.00E+04
2.50E+04
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
107
distance along the subsurface
depth (in)
0
0.00E+00
-0.002
-0.016
-0.018
-0.02
τ xz psi
Sxz analytical
Sxz numerical coarse mesh
Figure 6-36 Comparison of shear stresses in the XZ-direction for orientation 4
Sxz numerical fine mesh
108
The stress contour plots produced by analytical and FEM solutions in the case of
normal traction force for the case 1 are found to be in good agreement. As the tangential
traction force is zero the plots are symmetric about the vertical Y-axis.
The FEM results are evaluated for two different meshes for all the stresses and
compared with the Lekhnitskii’s analytical solution. First the results are plotted for case 1
where δ = γ = θ = 00. The second comparison is for the case 4 which is δ = γ = 10.610 and
θ = 400. The first mesh size is coarse and the second one is more refined for both the
cases. As seen in all the plots of stresses as a function of subsurface depth the results for
the refined mesh are closer to the analytical solution as compared to the coarse mesh
which shows that FEM results are approximately close and can be more accurate if the
mesh is further refined. The mesh is refined by dividing the region around the contact
with more number of nodes.
The stresses in the FEM solution are compared with the analytical stress values as
a function of subsurface depth.
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
The analysis procedure developed is used to examine subsurface stresses in
contact between a cylindrical isotropic indenter and a single crystal anisotropic substrate.
Subsurface stresses are studied for various crystallographic orientations. The resolved
shear stresses on the twelve primary octahedral planes [(111) <110>] are computed in the
vicinity of the contact, to predict evaluation of slip systems in the contact region.
Subsurface stresses are also computed using FEA, for a few cases, for comparison
purposes.
The results are applicable to the load of 4080lbs only as the contact area varies
non-linearly with load. This load value was used to get the contact width of 0.02in and
the maximum contact pressure of 260ksi on the anisotropic body in contact that has some
practical applications.
Conclusions
•
The analytical solution presented for the calculation of subsurface stresses in
anisotropic solids is shown to be an effective and efficient procedure for evaluating
subsurface stresses in anisotropic cylindrical contacts.
•
The resolved shear stresses calculated from the subsurface stresses in FCC single
crystal substrate provides a clear explanation of active slip planes and sectors for the
load used.
•
Slip sectors determined by the resolved shear stresses are not constant for the same
orientation, but depend on the value of the applied load and the yield stress (CRSS)
of the material.
109
110
•
The plots for resolved shear stresses as a function of radial and angular components
under the surface of the body, considering the normal and tangential traction force,
are not symmetrical for anisotropic solids.
•
The results of ANSYS finite element model used are comparable with the results of
the analytical solution.
•
Mesh refinement is needed in the FEM model for more accurate results.
•
The analytical approach is a fast and efficient technique as compared to the FEM
approach.
Recommendations for Future Work
The anisotropic elastic models used in this study are models that predict the slip
accurately, and can be further used to analyze the fatigue behavior with the help of
resolved shear stresses. The RSS values are calculated for the elastic region in this study.
The dominant slip system in the elastic region is only one of the parameters to predict
fatigue failure. Fatigue and fracture mechanics have to deal with the plastic models in
order to predict failure properly. In the plastic region failure occurs due to many
parameters such as creep, strain hardening, temperature effects, etc. Thus, analyzing the
stresses in the plastic regime is an important area to be researched in the future to predict
actual causes of failures in materials.
The study takes into account the bodies in contact that are non-conforming and
where the pressure profile due to contact is parabolic (like a Hertzian model) under a
normal load of 4080lbs. The results can be studied under various loads as contact area
changes non-linearly with the applied normal loads.
The tangential traction force in this study is considered to be in one direction
only. It is important to consider the reversal of loads as in fatigue loading. In the case of
reversal loading the effective resolved shear stress values on the all the twelve primary
111
octahedral slip planes at a point would be the difference between the values of the
resolved shear stresses in both the directions at that point.
APPENDIX
SAMPLE COORDINATE TRANSFORMATION WITH ACCURACY CHECKS
Sample Coordinate Transformation
x'
y'
z'
cos ( ψ 1) 0 −sin ( ψ 1) x
1
0
0
⋅ y
sin ( ψ 1) 0 cos ( ψ 1) z
x''
y''
z''
α 1 β 1 γ 1 xo
α 2 β 2 γ 2 ⋅ y o
α β γ z
3 3 3 o
α1 β 1 γ1
α2 β 2 γ2
α β γ
3 3 3
α 1 β 1 γ 1
α 2 β 2 γ 2
α β γ
3 3 3
x''
y''
z''
0
0
1
x'
0 cos ( ψ 2) sin ( ψ 2) ⋅ y'
0 −sin ( ψ 2) cos ( ψ 2) z'
0
0
1
cos ( ψ 1) 0 −sin ( ψ 1)
1
0
0 cos ( ψ 2) sin ( ψ 2) ⋅ 0
0 −sin ( ψ 2) cos ( ψ 2) sin ( ψ 1) 0 cos ( ψ 1)
cos ( ψ 1)
0
−sin ( ψ 1)
sin ( ψ 2) ⋅ sin ( ψ 1) cos ( ψ 2) sin ( ψ 2) ⋅ cos ( ψ 1)
cos ( ψ ) ⋅ sin ( ψ ) −sin ( ψ ) cos ( ψ ) ⋅ cos ( ψ )
2
1
2
2
1
y [010]
4 ψ1
2
x [100]
ψ2
z [001]
z″, Load direction [214]
112
113
Fig.App-1 Two step coordinate transformation
ψ 1 := atan
1
ψ 2 := atan
1
2
2⋅ 5
ψ 1 = 0.464
ψ 2 = 0.22
cos ( ψ 1)
0
−sin ( ψ 1)
α 1 β 1 γ 1
α 2 β 2 γ 2 := sin ( ψ 2) ⋅ sin ( ψ 1) cos ( ψ 2) sin ( ψ 2) ⋅ cos ( ψ 1)
α β γ
3 3 3 cos ( ψ 2) ⋅ sin ( ψ 1) −sin ( ψ 2) cos ( ψ 2) ⋅ cos ( ψ 1)
α 1 β 1 γ 1 0.894 0 −0.447
α 2 β 2 γ 2 = 0.098 0.976 0.195
α β γ
0.436 −0.218 0.873
3 3 3
Checks for Accuracy
All of the below should be equal to zero:
α1α2 + β1β2 + γ1γ2 = 0 α1α3 + β1β3 + γ1γ3 = 0
α1β1 + α2β2 + α3β3 = 0
α1γ1 + α2γ2 + α3γ3 = 0
α3α2 + β3β2 + γ3γ2 = 0
β1γ1 + β2γ2 + β3γ3 = 0
All of the below should be equal to one:
α12 + β12 + γ12 = 1
α22 + β22 + γ22 = 1
α32 + β32 + γ32 = 1
α12 + α22 + α32 = 1
β12 + β22 + β32 = 1
γ12 + γ22 + γ32 = 1
All the checks are valid and confirm the accuracy of the transformation
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BIOGRAPHICAL SKETCH
Sangeet Subhash Srivastava was born at Sewagram, Wardha, in India on April 10,
1978. He graduated from the Nagpur University with his B.E. in mechanical engineering
in July 2000. Then he joined the University of Florida in the Mechanical Engineering
Department in August 2000 to pursue his graduate studies.
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